Apparatus and method for providing higher radix redundant digit lookup tables for recoding and compressing function values

ABSTRACT

An apparatus and method are disclosed for providing higher radix redundant digit lookup tables for digital lookup table circuits. A compressed direct lookup table unit accesses a redundant digits lookup table that is capable of providing a high order part and a low order part that can be directly concatenated to form an output numeric value. The redundant digits lookup table of the invention is structured so that no output overflow exceptions are created. A redundant digits lookup table recoder capable of providing recorded output values directly to partial product generators of a multiplier unit is also disclosed.

CROSS REFERENCE TO RELATED APPLICATION

The present invention is related to that disclosed in the following U.S.Non-Provisional Patent Application:

-   -   Ser. No. 10/108,251 filed concurrently herewith, entitled        “APPARATUS AND METHOD FOR MINIMIZING ACCUMULATED ROUNDING ERRORS        IN COEFFICIENT VALUES IN A LOOKUP TABLE FOR INTERPOLATING        POLYNOMIALS.”

The above patent application is commonly assigned to the assignee of thepresent invention. The disclosures in this related patent applicationare hereby incorporated by reference for all purposes as if fully setforth herein.

FIELD OF THE INVENTION

The present invention relates generally to the field of computertechnology, and more particularly to an arithmetic unit of a computer.The present invention provides an improved apparatus and method forproviding lookup tables for function values.

BACKGROUND OF THE INVENTION

In binary computing devices hardware direct lookup tables are typicallyemployed for function evaluation and for reciprocal and root reciprocalseed values for division and square root procedures. For direct tablelookup of a function of a normalized “p-bit” argument 1≦x=1.b₁b₂ . . .b_(i)b_(i+1) . . . b_(p−2), the “i” leading bits b₁b₂ . . . b_(i)provide an index to a table yielding “j” output bits that determine theapproximate function value.

Two major disadvantages of prior art hardware direct lookup tables are:

-   -   (1) A prior art hardware direct lookup table generally contains        considerable repetition of common leading portions of output        values. This results in the table being considered prohibitively        large for certain applications, particularly when the        application requires the number of input bits “i” to be equal or        greater than the number of output bits “j”.    -   (2) For direct lookup of functions that change slowly in some        portions of the argument domain and then change very rapidly in        other portions of the argument domain, the effective accuracy        and amount of output digit repetition that is generated varies        considerably over the range of output values.

To provide greater function accuracy and to avoid an excessive size of acomparable direct lookup table, certain types of prior art table lookupsystems employing multiple table lookups and arithmetic combiningoperations have been devised. These table lookup systems includebipartite tables such as those described in U.S. Pat. No. 5,862,059.Another type of table lookup system uses exponential/logarithm tablessuch as those described in U.S. Pat. No. 5,197,024. Other types of tablelookup systems are based on well known multiplicative interpolationsystems.

Some of the disadvantages of these prior art types of hardware tablelookup systems for function approximation include:

-   -   (1) The table lookups and related computations can require        several cycles of latency on the critical path of the overall        function evaluation procedure.    -   (2) The table lookup system can require supplemental dedicated        hardware to effect the implicit table size reduction, and    -   (3) A table lookup assisted computation generally provides        alternative approximate function values differing from those of        a comparably accurate direct table lookup. Therefore the table        lookup assisted computation procedure is a “lossy compression”        procedure.

It should be recognized that even though a lookup table providesapproximate values, there are many reasons why table compression shouldstill be a lossless compression rather than just a methodology toprovide values of roughly the same accuracy as the initial direct lookuptable values. These reasons include the issues of adherence tostandards, the maintenance of legacy systems behavior, and the facilityfor reproducible behavior for verification and testing.

A prior art direct table lookup procedure that incorporates losslesscompression techniques to avoid excessive repetition of common leadingdigits of successive output values enumerated in the table is a loworder digit accuracy refinement method used in published decimal lookuptables for function approximation. For example, consider a decimallogarithm table of the type printed in the book “Mathematical Tablesfrom the Handbook of Chemistry and Physics” published for many years bythe Chemical Rubber Publishing Company of Cleveland, Ohio.

A published lookup table such as a five (5) digit index decimallogarithm table is an example of a decimal based direct lookup table.Each decimal input value indexes an output with a corresponding numberof output decimal digits (plus perhaps a fixed number of extra guarddigits to provide more output accuracy). Thus the five (5) digit inputvalue N=11502 can index an eight (8) digit output value 4.0607734. Theoutput value 4.0607734 is the rounded decimal logarithm of the inputvalue N=11502. Similarly, the five (5) digit input value N=11509 canindex the output value 4.0610376. The output value 4.0610376 is therounded decimal logarithm of the input value N=11509. Table compressionin the printed table may be obtained by employing the low order decimalinput index digit as a refined value selector.

For purposes of explaining the concept of table compression the inputindex can be partitioned into high and low order indices. Let theexpression 4|1 denote an “input partition” with one low order refinementdigit. The digit “4” in the expression 4|1 denotes four (4) high orderdigits that index a line of the table where the outputs are given in apartitioned form. The digit “1” in the expression 4|1 denotes one (1)low order digit that indexes one column in the table.

Let the expression 4|4 denote an “output partition” having a high orderoutput part of four (4) digits and a low order output part of four (4)digits. Line compression is obtained by having a common high order partfollowed by a sequence of ten (10) low order parts each indexed by thevalue of the one (1) low order refinement digit.

For example, consider the following two lines from a five (5) digitindex decimal logarithm direct lookup table published by Chemical RubberPublishing Company.

TABLE 1 N 0 1 2 8 9 1150 060 6978 7356 7734 . . . 9999 *0376 1151 0610753 1131 1508 . . . 3771  4148

The output of this logarithm table provides a rounded eight (8) digitdecimal logarithm partitioned in a 4|4 output partition. The leadingfour (4) digits of the logarithm value (i.e., d₀. d₁ d₂ d₃) are given bythe value d₀ and the common three leading fraction digits d₁ d₂ d₃. Thedigit d₀=4 is assumed to be known to the user of the table from the factthat N is a five (5) digit number. In the two lines of the logarithmtable shown in TABLE ONE the common three leading fraction digits d₁ d₂d₃ are either “060” or “061.” The remaining four (4) low order digits d₄d₅ d₆ d₇ refining the output value are obtained from the entries in thecolumns of the table with the low order refinement digit of the inputselecting the particular column.

The table lookup process for looking up the logarithm of the input valueN=11501 involves (a) accessing the table values in line “1150” and (b)concatenating the three digits “060” to the four (4) digit low orderpart “7356” from column “1” and (c) concatenating the fraction to thed₀=4 value. The result is the output logarithm value of 4.0607356.

The table lookup process for looking up the logarithm of the input valueN=11509 involves (a) accessing the low order table values in line “1150”and (b) recognizing the low order selection “overflow” indicator (i.e.,the asterisk) and (c) concatenating the three digits “061” in lire“1151” to the four (4) digit low order part “0376” from column “9”, and(d) concatenating the fraction to the d₀=4 value. The result is theoutput logarithm value of 4.0610376. The asterisk placed on the value“0376” in column “9” indicates that the three digits “061” from line“1151” are to be used instead of the three digits “060” from the line“1150.”

The input value N=11509 creates an “overflow” exception condition in thelow order output part that requires the low order output part “0376” tobe concatenated to the high order output part incremented by a carry. Inthis case the high order output part on line “1150” is “060.” When it isincremented by a carry the value becomes “061” which is the high orderpart of the next line “1151.” The output “overflow” exception conditionis denoted in the lookup table by a leading asterisk included in the loworder output part.

The output “overflow” exception condition described above for apublished decimal lookup table may also be employed in an analogousbinary lookup table (i.e., a table in which the output numbers areprinted in binary form). The mathematical principles that are involvedare the same. Binary lookup tables are not usually provided in a printedtable. This is because the user of a binary lookup table is almostalways an “arithmetic logic unit” (ALU) of a computer. Binary lookuptables are usually provided in a computer “read only memory” (ROM). AnALU directly accesses the ROM binary lookup table to obtain desiredoutput values from the table.

Some disadvantages of these prior art low order digit refinement methodsare as follows:

-   -   (1) Selective concatenation of high and low order digits has a        delay due to the dependency of the high order digit selection on        the output of the low order digit selection. That is, it must        first be determined whether the asterisk symbol that may be        output with the low order digits is present or absent.    -   (2) Neither the high order digits nor the low order digits are        available from the initial lookup of the line.    -   (3) A variety of formatting options for adapting the range of        output values are seamlessly employed to achieve greater        compression and to reduce the visual clutter on the printed        page. An example of a printed page is Page 45 of the book        Standard Mathematical Tables, Fourteenth Edition, published by        The Chemical Rubber Company of Cleveland, Ohio, shown in        Appendix 1. While the reader seamlessly reads the published        table, an equivalent hardware implementation can require        excessive branching logic and delay in a similar hardware table        system that comprises a plurality of ROM tables with a complex        selection mechanism.

Accordingly, a need has arisen for a method of compression in a directlookup table where the output is available with a minimum of conditionalpost-lookup table logic. A further need has arisen to obtain a directlookup table compression that is lossless, in order to preserve theintegrity of the table lookup values. A further need has arisen for amethod for designing the partition of a direct lookup table into aplurality of lookup tables, together with a simple selection mechanism,to enable a desired lookup table of a plurality of lookup tables inorder to obtain a desired lookup table value while minimizing powerusage and achieving a higher level of lossless compression.

A further need has arisen for an improved apparatus and method forproviding a direct lookup table in which the digit values in the directlookup table are encoded in bits with the bits allocated to parts of theoutput so as to minimize the total number of bits required to expressthe values in the direct lookup table thereby achieving greatercompression.

SUMMARY OF THE INVENTION

The present invention is directed to an apparatus and method forproviding a direct table lookup system for function evaluation in whichparts comprising consecutive digit sub-sequences of the table lookupfunction value in a redundant digit format are obtained from a pluralityof direct lookup tables. The table lookup function value in theredundant digit format is obtained from the concatenation of the partsthereby providing a lookup table system in which output “overflow”exception conditions do not occur.

A technical advantage of the present invention inheres in the fact thata redundant digit format of the parts allows the parts to be provided bytables that are substantially compressed compared to an equivalentsingle direct lookup table. The redundant digit format table lookupvalue is the same value as that from the single direct lookup tableresulting in a “lossless” compression.

An additional technical advantage of the present invention inheres inthe fact that the table lookup function value redundant digit format maycontain nonredundant as well as redundant digits where the binaryencoding of the redundant digits may be split so that some bits of theencoding occur in each of two successive parts of the table lookupfunction value redundant digit format. The ability of the format toinclude nonredundant as well as redundant digits allows furtherreduction in a table size obtaining greater lossless compression.

Another technical advantage inheres in the present method of theinvention for partitioning a large direct lookup table into a pluralityof smaller direct lookup tables. The method of the present inventionprovides partitions that allow for convenient selection of the smallertables that are to be enabled so as to both reduce power and obtaingreater lossless compression.

It is an object of the present invention to provide an apparatus andmethod for providing a lookup table system in which output “overflow”exception conditions do not occur.

It is another object of the present invention to provide an apparatusand method for providing a lookup table in which the values in thelookup table are encoded to minimize the number of bits required toexpress the values in the lookup table.

It is also an object of the present invention to provide a direct tablelookup system for function evaluation in which parts comprisingconsecutive digit sub-sequences of the table lookup function value in aredundant digit format are obtained from a plurality of direct lookuptables and concatenated to obtain a table lookup function value.

It is another object of the present invention to provide a direct tablelockup system for function evaluation in which the redundant digitformat of the parts are substantially compressed compared to a singledirect lookup table.

It is also an object of the present invention to provide a direct tablelockup system for function evaluation in which a table lookup functionvalue in a redundant digit format may contain nonredundant as well asredundant digits where the binary encoding of the redundant digits maybe split so that some bits of the encoding occur in each of twosuccessive parts of the table lookup function value redundant digitformat.

The foregoing has outlined rather broadly the features and technicaladvantages of the present invention so that those skilled in the art maybetter understand the Detailed Description of the Invention thatfollows. Additional features and advantages of the invention will bedescribed hereinafter that form the subject matter of the claims of theinvention. Those skilled in the art should appreciate that they mayreadily use the conception and the specific embodiment disclosed as abasis for modifying or designing other structures for carrying out thesame purposes of the present invention. Those skilled in the art shouldalso realize that such equivalent constructions do not depart from thespirit and scope of the invention in its broadest form.

Before undertaking the Detailed Description of the Invention, it may beadvantageous to set forth definitions of certain words and phrases usedthroughout this patent document: The terms “include” and “comprise” andderivatives thereof, mean inclusion without limitation, the term “or” isinclusive, meaning “and/or”; the phrases “associated with” and“associated therewith,” as well as derivatives thereof, may mean toinclude, be included within, interconnect with, contain, be containedwithin, connect to or with, couple to or with, be communicable with,cooperate with, interleave, juxtapose, be proximate to, to bound to orwith, have, have a property of, or the like; and the term “controller,”“processor,” or “apparatus” means any device, system or part thereofthat controls at least one operation. Such a device may be implementedin hardware, firmware or software, or some combination of at least twoof the same. It should be noted that the functionality associated withany particular controller may be centralized or distributed, whetherlocally or remotely. Definitions for certain words and phrases areprovided throughout this patent document. Those of ordinary skill shouldunderstand that in many instances (if not in most instances), suchdefinitions apply to prior uses, as well as to future uses, of suchdefined words and phrases.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention, and theadvantages thereof, reference is now made to the following descriptionstaking in conjunction with the accompanying drawings, wherein likenumbers designate like objects, and in which;

FIG. 1 illustrates a block diagram of a portion of a central processingunit showing a prior art arithmetic logic unit comprising a lookup tableand an arithmetic unit.

FIG. 2 illustrates a block diagram of a prior art arithmetic logic unitcomprising a lookup table, a multiplier recoder, and a multiplier unit;

FIG. 3 illustrates three portions of a reciprocal binary lookup tableaccording to one advantageous embodiment of the present invention;

FIG. 4 illustrates a block diagram of an advantageous embodiment of acompressed direct lookup table of the present invention;

FIG. 5 illustrates a block diagram of an alternate advantageousembodiment of a compressed direct lookup table of the present invention;

FIG. 6 illustrates a block diagram of an alternate advantageousembodiment of a compressed direct lookup table of the present inventionthat incorporates a Booth “radix 4” encoding format for the compresseddirect lookup table;

FIG. 7 illustrates a block diagram of another alternate advantageousembodiment that is capable of obtaining a lower power implementation ofa compressed direct lookup table of the present invention; and

FIG. 8 illustrates a flow chart showing the steps of one advantageousembodiment of a method of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1 through 8, discussed below, and the various embodiments used todescribe the principles of the present invention in this patent documentare by way of illustration only and should not be construed in any wayto limit the scope of the invention. Those skilled in the art willunderstand that the principles of the present invention may beimplemented in any suitably arranged look up table unit in a centralprocessing unit of a computer system.

FIG. 1 illustrates a block diagram of a portion of a central processingunit 100 showing a prior art arithmetic logic unit (ALU) 110. Arithmeticlogic unit 110 comprises a lookup table (LUT) 120 and an arithmetic unit130. Lookup table 120 receives data in the form of “i” input bits.Lookup table 120 outputs data to arithmetic unit 130 in the form of “j”output bits in accordance with principles that are well known in theprior art. Arithmetic unit 130 may comprise a multiplier unit, an adderunit, a microprogram storage unit, or other type of computational unit.

FIG. 2 illustrates a block diagram of a prior art arithmetic logic unit(ALU) 200 comprising lookup table 210, multiplier recoder 220, andmultiplier unit 230. Lookup table 210 receives data in the form of “i”input bits. Lookup table 210 outputs data to multiplier recoder 220 inthe form of “j” output bits in accordance with principles that are wellknown in the prior art. Multiplier recoder 220 comprises a Booth “radix4” multiplier recoder. Also in accordance with principles that are wellknown in the prior art multiplier recoder 220 outputs to multiplier unit230 a number of bits equal to$3\quad\left\lceil \frac{\left( {j + 1} \right)}{2} \right\rceil$bits where the bottomless brackets denote taking the smallest integergreater than or equal to the expression within the bottomless brackets.

FIG. 3 illustrates three portions of a reciprocal binary lookup table300 according to one advantageous embodiment of the invention. The inputto binary lookup table 300 is provided in the form of thirteen (13)binary bits that represent an input number “x”. As will be more fullydescribed, the output of binary lookup table 300 is in the form of seven(7) “radix 4” digits that represent an approximate reciprocal of theinput number “x” (i.e., the value of “(1/x)”).

The inputs are normalized to the standard binary interval [1, 2) withtwelve (12) fraction bits (i.e., 1.b₁b₂ b₃ . . . b₁₁ b₁₂). The standardbinary interval [1, 2) comprises binary values that are less than two(2) and that are greater than or equal to one (1). Parentheses ( )indicate exclusive bounds and brackets [ ] indicate inclusive bounds.

If all of the lines in reciprocal binary lookup table 300 were shown inFIG. 3 there would be a total of five hundred twelve (512) lines. Acomplete version of reciprocal binary lookup table 300 with all fivehundred twelve (512) lines is set forth in Appendix “A” for referencepurposes. The leading nine (9) fraction bits (i.e., b₁ b₂ b₃ . . . b₈b₉)select one of the five hundred twelve (512) lines of the complete table.FIG. 3 shows the first eight (8) lines, the middle sixteen (16) lines,and the final eight (8) lines of reciprocal binary lookup table 300. Thelow order three bits of the index (i.e., b₁₀b₁₁b₁₂) select one of eight(8) columns providing the low order digit part to be concatenated to thehigh order digit part at the beginning of that line.

The output values in reciprocal binary lookup table 300 are determinedas follows. Consider an input 1.b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂=i2 ⁻¹² thatdenotes the interval [i2 ⁻¹², (i+1)2⁻¹²) where the value of “i” is inthe range 1024≦i≦2047. The reciprocal of the midpoint falls in theinterval (1½, 1), and is computed and first “rounded-to-nearest” totwelve (12) binary places beyond the leading unit. The result is R$\left( {2^{12}/\left( {i + \frac{1}{2}} \right)} \right) = {0.1\quad b_{1}^{\prime}\quad b_{2}^{\prime}\quad b_{3}^{\prime}\quad b_{4}^{\prime}\quad b_{5}^{\prime}\quad b_{5}^{\prime}\quad b_{6}^{\prime}\quad b_{7}^{\prime}\quad b_{8}^{\prime}\quad b_{9}^{\prime}\quad b_{10}^{\prime}\quad b_{11}^{\prime}\quad{b_{12}^{\prime}.}}$For displaying the output entry value in the table of FIG. 3 the resultis multiplied by two (2) to normalize the value into the standard binaryinterval [1, 2). The value is then converted to a “radix 4”representation with all of the digits selected from the redundant digitset {−2, −1, 0, 1, 2}. This yields a seven (7) digit “radix 4” numberd₀.d₁d₂d₃d₄d₅d₆ that is identical in value to the thirteen (13) bitnormalized binary value 1.b′₁b′₂b′₃b′₄b′₅b′₆b′₇b′₈b′₉b′₁₀b ₁₁b′₁₂.

In many applications a set of “radix 4” digits is made up of the“standard” set of digits {0, 1, 2, 3}. In binary computer applications,such as multiplication by a digit, the number three (3) is not aconvenient digit to use. This is because in order to multiply a numberby three (3) the number must be shifted once (to multiply the number bytwo in the base two) and the result added to the number (to obtain aresult that equals three times the number). In contrast, to multiply anumber by two (2) only requires one shift operation. To multiply anumber by negative one (−1) only requires one complement operation, andto multiple a number by negative two (−2) requires only both acomplement operation and a shift operation. Thus multiplication by anyof the digits {−2, −1, 0, 1, 2} is simpler and involves less delay thanmultiplication by 3.

It is therefore more convenient to use a range of “radix 4” digits thatdoes not include the digit three. In lookup table 300 the range of“radix 4” digits used is the redundant digit set {−2, −1, 0, 1, 2},which is often termed the Booth “radix 4” digit set.

Each line of reciprocal binary lookup table 300 provides a common four(4) digit high order part d₀.d₁d₂d₃ with a selected low order partd₄d₅d₆ concatenated to generate the output entry value of the lookuptable. Bars placed over the numbers one (1) and two (2) in the table areused to denote negative values. That is, “bar 1” (shown as {overscore(1)}) is equal to “minus 1” (−1) and “bar 2” (shown as 2) is equal to“minus 2” (−2) in the printed digit strings.

For example, the table input index 1.000 000 110 010 employs thenumerals 1.000 000 110 to determine the line with high order output2.00{overscore (2)} and the low order input bits 010 determine thecolumn giving the low order output 21{overscore (1)}0. Concatenating thetwo parts yields the output value 2.00{overscore (2)} 2{overscore (1)}0.

A reciprocal table with normalized inputs 1≦1. b₁b₂b₃ . . . b_(i)<2 andnormalized outputs 1≦1. b′₁b′₂b′₃ . . . b′_(j)<2 is referred to as an“i-bits-in, j-bits-out” reciprocal table. The leading unit bit isassumed in the normalization. Reciprocal binary lookup table 300 thenrepresents a “twelve (12) bits in, twelve (12) bits out” reciprocaltable with the output converted (exactly) to the normalized seven (7)digit “radix 4” output d₀.d₁d₂d₃d₄d₅d₆ Note that the leading outputdigit can be restricted to have the value two (d₀=2) for inputs in theinterval [1, 1½), and restricted to have the value one (d₀=1) for inputsin the interval [1½, 2). This output digit value is available from theinput bit b₁ and therefore does not need to be stored. The “radix 4”reciprocal binary lookup table 300 in FIG. 3 may be referred to as a “2kbits in, k “radix 4” digits out” table (where the value of k is equal tosix (6)).

To achieve greater compression it is desirable to have the common highorder digit string large and the low order digit string small. To avoidthe output “overflow” exception condition encountered in traditionallookup tables it is sufficient that the leading low order digit in allentries across a line has at most two successive values. Note that thefirst line of reciprocal binary lookup table 300 has leading low orderdigit values “zero” (“0”) and “minus one” (“−1”), the second line hasvalues “minus one” (“−1”) and “minus two” (“−2”), the third line hasvalues “two” (“2”) and “one” (“1”), and the fourth line has values “one”(“1”) and “zero” (“0”).

With a change of at most one unit in the leading low order digit valueacross a line, the redundancy in the digit set, specifically, thefacility to choose either “2” or “−2” as the low order leading digit,provides that a common high order part may always be provided applicableto the whole line. This allows a high order part to be selected and anyof a corresponding set of low order parts to be selected that may alwaysbe concatenated without any overflow exception. The hardware design of acompressed table is greatly simplified if hardware does not need to beprovided to handle overflow exceptions.

The recoded output obtained from the tables at the digit level has beendescribed using the five “radix 4′ values {−2, −1, 0, 1, 2}. It isnecessary further to consider the encoding of these five values in bits.Typical multiplier encodings employ a sign bit and two magnitude bits toprovide the digit value as input to a partial product generator (“PPG”)(not shown). The encoding may be stored directly in a table. However,this will increase the table size and thereby reduce the netcompression. In exemplary look up table 300 each line of the table hasentries that total twenty seven (27) “radix 4” digits. With a three (3)bit encoding per digit the line width is eight one (81) bits. The linewidth for standard binary output would have eight (8) words of twelve(12) bits each for a total of ninety six (96) bits. In this case somecompression is achieved together with the convenience of an output digitencoding that is suitable for direct input to a PPG of a multiplierunit.

Another advantage of the present invention is that the output digitencoding may be refined to improve the compression while representingthe same redundant digit values.

For example, note that all the output digits except the assumed leadingdigit d₀ and except the leading digit of the low order part (d₄ in table300) have values in the limited range {−2, −1, 0, 1}. This feature isalways possible when redundancy is only employed in selecting theleading digit of the low order part to insure against overflow from lowto high order part across a line. This means that all but one of the“radix 4” digits can be encoded with only two (2) bits. For example, thedigits “00” denote the value “0”, the digits “01” denote the value “+1”,the digits “10” denote the value “−2”, and the digits “11” denote thevalue “−1”, as in a two (2) bit “2's complement” encoding of the values−2, −1, 0, 1. This allows that the twenty seven (27) “radix 4” digits oneach line of table 300 to be stored with eighteen (18) digits having two(2) bit encodings and only eight (8) digits having three (3) bitencodings, for a line width of only sixty two (62) bits.

A digit that represents a digit value in the full redundant range of{−2, 1, 0, 1, 2} can be encoded with three (3) bits as shown in TABLETWO below. A sign bit and a two (2) bit “carry-save” value is employedto encode the digit magnitude.

TABLE 2 Two encodings of the digit range [−2, 2]. A Booth “Radix 4”Compression Maximum Digit Encoding Encoding −2 110 111 −1 101 101 or 1100 100 or 000 000 or 100 1 001 001 or 010 2 010 011

If the compression maximum encoding of TABLE TWO comprising bits b1, b2,and b3 is used for the leading digit of the low order part, note thatthe first two (2) bits, b1 and b2, of that digit may be chosen to remainconstant across any line of output, and may therefore be attached to thehigh order digit encoded part. Only a single bit b3 of this digit needbe replicated with each of the low order parts. This results in anoutput line width comprising six (6) bits for the three (3) high order“radix 4′ digit encodings, and two (2) bits for the non-changing part ofthe leading low order digit encoding, and eight (8) sets of five (5)bits each for completing the low order part encodings. This yields fortyeight (48) bits per line, a fifty percent (50%) reduction over theninety six (96) bits of the standard eight (8) bit by twelve (12) bitoutput line.

This advantageous encoding of the present invention provides a greatercompression with the output still available by concatenation of theparts. One more step could then be applied to convert the encoding fromthe alternate three (3) bit form of TABLE TWO or of the two (2) bit “2'scomplement” form into the desired Booth Radix 4” encoding of TABLE TWOof the digits {−2, −1, 0, 1, 2} for input to a partial product generator(PPG) of a multiplier unit. Note that for a multiplier recoding unitthat encodes with a sign bit, a magnitude two (“2”) select bit, and amagnitude one (“1”) select bit, as in the Booth “Radix 4” encoding ofTABLE TWO, the converted encoding of table 300 in a single logic levelis achieved by having the sign bit equal to b₁, by having the select one(“1”) bit equal to b₂ XOR b₃, and by having the select two (“2”) bitdetermined by the value of b₂ AND b₃.

FIG. 4 illustrates a block diagram of an advantageous embodiment of acompressed direct lookup table unit 400 of the present invention.Compressed direct lookup table unit 400 receives into input latch 410“p” input bits in the form, of “1.b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂b₁₃ . . .b_(p−1).” Input latch 410 provides twelve (12) “leading” input bitsb₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂ to lookup table unit 420.

In particular, input latch 410 provides the leading nine (9) input bitsb₁b₂b₃b₄b₅b₆b₇b₈b₉ to a redundant digits lookup table unit 430 of lookuptable unit 420. Redundant digits lookup table unit 430 comprises thehigh order output parts of reciprocal binary lookup table 300 along withthe two (2) common bits of the encoding of the eight (8) low orderterms. Redundant digits lookup table unit 430 obtains the high orderpart of the output value. Input latch 410 also provides the twelve (12)leading input bits b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂ to a redundant digitslookup table unit 440 of lookup table unit 420. Redundant digits lookuptable unit 440 obtains the low order part of the output value.

Redundant digits lookup table unit 430 sends to result latch 450 nine(9) output bits that encode the “radix 4” digits of the high order partof the output, where two (2) of these nine (9) bits encode a common partof the leading redundant digit encoding of the low order part. Six (6)bits encode the three (3) “radix 4” digits d₁ d₂ d₃ in nonredundant formusing two (2) bits for each digit, and the first input bit b₁ is alsooutput as the encoding of d₀. Thus redundant digits lookup table unit430 may be viewed as a “nine (9) bits in, nine (9) bits out” tableproviding the high order part of the output in a compressed encoding.

Redundant digits lookup table unit 440 uses the leading twelve (12)input bits b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂ to obtain the appropriate loworder digits part which can be found on a selected line of the fivehundred twelve (512) lines of reciprocal binary lookup table 300. Inparticular, the leading nine (9) input bits b₁b₂b₃b₄b₅b₆b₇b₈b₉ determinethe appropriate line of the five hundred twelve (512) lines ofreciprocal binary lookup table 300 and the three (3) input bitsb₁₀b₁₁b₁₂ designate the appropriate low order digits part in theselected line. Redundant digits lookup table unit 440 is constructed tohave an encoding of these three (3) digits and sends to result latch 450five (5) output bits that encode these three (3) “radix 4” digits of thelow order part of the output. The encoding comprises two (2) bits eachfor the nonredundant digits d₅ and d₆ and the lowest bit of the three(3) bit compression maximum encoding of digit d₄. Thus Redundant digitslockup table unit 440 may be viewed as a “twelve (12) bits in, five (5)bits out”table providing the low order part of the output in thecompressed encoding.

The concatenation of the high and low order parts provides the fourteen(14) bit encoding of the seven (7) digit “radix 4” output in thecompressed encoding.

FIG. 5 illustrates a block diagram of an alternate advantageousembodiment of a compressed direct lookup table unit 500 of the presentinvention. Compressed direct lookup table unit 500 receives into inputlatch 510 “p” input bits in the form of “1.b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂b₁₃ . . . b_(p−1).” Input latch 510 provides twelve (12)“leading” input bits b₁b₂b₃b₄b₅b₆b₇b₈b₉b₁₀b₁₁b₁₂ to lookup table unit520.

In particular, input latch 510 provides the leading nine (9) input bitsb₁b₂b₃b₄b₅b₆b₇b₈b₉ to a redundant digits lookup table unit 530 of lookuptable unit 520. Redundant digits lookup table unit 530 comprisesreciprocal binary lookup table 300. Redundant digits lookup table unit530 obtains the high order part of the output value. Input latch 510also provides three (3) low order input bits b₁₀b₁₁b₁₂ as control bitsto a low order part multiplexer 540 of lookup table unit 520. Low orderpart multiplexer 540 identifies the correct low order part of the outputvalue. The input bits b₁₀b₁₁b₁₂ are available to establish the paththrough the low order part multiplexer 540 while lookup occurs inredundant digits lookup table unit 530 so that there is little delay inselecting the desired low order part to be passed to result latch 550.

Redundant digits lookup table unit 530 uses the leading nine (9) inputbits b₁b₂b₃b₄b₅b₆b₇b₈b₉ to find the appropriate line of the five hundredtwelve (512) lines of reciprocal binary lookup table 300. Redundantdigits lookup table unit 530 sends to result latch 550 nine (9) outputbits that represent the “radix 4” digits of the high order part of theoutput along with the two (2) bits of the common part of the encoding ofthe leading digit of the low order part.

Redundant digits lookup table unit 530 also sends forty (40) bits to loworder part multiplexer 540. The first five (5) bits of these forty (40)bits encode a first set of three (3) “radix 4” digits. This first set ofthree (3) “radix 4” digits encode a first one of the eight (8) low orderoutput values in the selected line of table 300.

The second five (5) bits of these forty (40) bits encode a second set ofthree (3) “radix 4” digits. This second set of three (3) radix 4” digitsrepresent a second one of the eight (8) low order output values in theselected line of table 300.

Similarly, the remaining six (6) sets of five (5) bits encoderespectively the remaining six (6) sets of three (3) “radix 4” digitsthat represent low order output values in the selected line of table300.

Low order part multiplexer 540 uses the three (3) low order input bitsb₁₀b₁₁b₁₂ from input latch 510 as control bits to select the appropriateset of three (3) “radix 4” digits from the eight (8) sets in theselected line of table 300. After the appropriate set of three (3)“radix 4” digits has been selected, low order part multiplexer 540 sendsto result latch 550 five (5) output bits that encode the “radix 4”digits of the low order part of the output.

Because reciprocal binary lookup table 300 uses a redundant digitrepresentation (here, a “radix 4” digit representation) there is nooutput “overflow” condition in the lines of table 300. Therefore, theoutput of redundant digits lookup table unit 530 and the output of loworder part multiplexer 540 may be directly concatenated in result latch550 to provide the desired output.

As an example, consider an input value of “1.011 111 010 101.” The nine(9) high order bits “011 111 010” indicate the third line in the secondportion of table 300 shown in FIG. 3. Redundant digits lookup table unit530 sends to result latch 550 nine (9) output bits that represent the“radix 4” digits of the high order part of the output (i.e., the “radix4” digits 2.{overscore (222)}).

Redundant digits lookup table unit 530 also sends forty (40) bits to loworder part multiplexer 540 that represent the eight (8) sets of three(3) “radix 4” digits in the selected line. The eight (8) sets are [000][00{overscore (1)}] [00{overscore (2)}] [0{overscore (1)}{overscore(1)}] [0{overscore (1)}0 ] [0{overscore (1)}{overscore (1)} ][0{overscore (1)}{overscore (2)}] [0{overscore (1)}{overscore (2)}]. Thebrackets are employed for clarity to indicate the grouping of the “radix4” digits. Low order part multiplexer 540 uses the three (3) low orderinput bits “101” from input latch 510 as control bits to select thegroup [0{overscore (1)}{overscore (1)}] from the group of eight (8)sets.

The output encoding of the digits 2.{overscore (222)} of redundantdigits lookup table unit 530 and the output encoding of the digits0{overscore (1)}{overscore (1)} of low order part multiplexer 540 aredirectly concatenated in result latch 550 to provide the desiredencoding of the output of 2.{overscore (2)}{overscore (2)}{overscore(2)}0{overscore (1)}{overscore (1)}.

The compressed direct lookup table unit 500 of the embodiment of theinvention shown in FIG. 5 comprises reciprocal binary lookup table 300.Reciprocal binary lookup table 300 is an illustrative example of aredundant digits lookup table for performing a function evaluation(here, obtaining a reciprocal of a number) It is understood that thepresent invention is not limited to a redundant digits lookup table forobtaining a reciprocal of a number. The invention is directed to anyredundant digits lookup table that is capable of performing a functionevaluation.

FIG. 6 illustrates a block diagram of an alternate advantageousembodiment of a compressed direct lookup table unit 600 of the presentinvention. In this alternate embodiment the redundant digits lookuptable unit 530 functions as a lookup table recoder unit 620. Lookuptable recoder unit 620 also comprises reciprocal binary lookup table300. Lookup table recorder unit 620 uses reciprocal binary lookup table300 to perform the lookup function performed by redundant digits lookuptable unit 530. In addition, lookup table recoder unit 620 recodes theoutput numeric values obtained from reciprocal binary lookup table 300.

In the embodiment shown in FIG. 6, lookup table recoder unit 620performs Booth “radix 4” recoding. The high order part of the recodedoutput of lockup table recoder unit 620 goes directly to partial productgenerators (PPG) 640 a through 640 e. Lookup table recoder unit 620provides three (3) bits of a recoded output value to each of PPG 640 a,PPG 640 b, PPG 640 c, and PPG 640 d. Lookup table recoder unit 620provides one (1) bit of a recoded output value to PPG 640 e.

Lookup table recoder unit 620 also provides sixty four (64) bits to loworder part multiplexer 630 that represent recoded versions of the eight(8) sets of three (3) “radix 4” digits. In the same manner as low orderpart multiplexer 540, low order part multiplexer 630 uses the three (3)low order input bits b₁₀b₁₁b₁₂ from input latch 610 to select theappropriate recoded set of three (3) “radix 4” digits from the recodedversions of the eight (8) sets in the selected line of table 300. Loworder part multiplexer 630 identifies the correct low order part of therecoded output value.

Because reciprocal binary lookup table 300 uses a redundant digitrepresentation (here, a “radix 4” digit representation) there is nooutput “overflow” condition in the lines of table 300. Therefore, theoutput of lookup table recoder unit 620 and the output of low order partmultiplexer 630 may be directly concatenated in partial productgenerators, 640 a through 640 g.

Low order part multiplexer 630 provides three (3) bits of a recodedoutput value to PPG 640 f and three (3) bits of a recoded output valueto PPG 640 g. Low order part multiplexer 630 provides two (2) bits of arecoded output value to PPG 640 e.

The outputs of partial product generators, 640 a through 640 g, are thenprovided to multiplier core 650 in accordance with well knownprinciples.

The compressed direct lookup table unit 600 of the embodiment of theinvention shown in FIG. 6 comprises reciprocal binary lookup table 300.Reciprocal binary lookup table 300 is an illustrative example of aredundant digits lookup table for performing a mathematical operation(here, obtaining a reciprocal of a number). It is understood that thepresent invention is not limited to a redundant digits lookup table forobtaining a reciprocal of a number. The invention is directed to anyredundant digits lookup table that is capable of providing a functionvalue.

FIG. 7 illustrates a block diagram of another alternate advantageousembodiment of a compressed direct lookup table unit 700 of the presentinvention. The embodiment of the invention shown in FIG. 7 comprises afirst redundant digits lookup table 730 and a second redundant digitslookup table 740. As will be more fully discussed, the use of tworedundant digit lookup tables allows greater compression and reducesoverall power consumption.

Compressed direct lookup table unit 700 comprises a read only memory(ROM) select unit 720 that is capable of receiving three (3) bits of theinput numeric value that is located in input latch 710. These three (3)bits represent the first three (3) leading bits of the input numericvalue (i.e., bits “b1” and “b2” and “b3”).

When read only memory (ROM) select unit 720 reads any bit combinationother than three (3) ones (“111”) then read only memory (ROM) selectunit 720 activates power in that portion of compressed direct lookuptable unit 700 that comprises first redundant digits lookup table unit730 and low order part multiplexer 750. Power is not activated in theportion of compressed direct lookup table unit 700 that comprises secondredundant digits lookup table unit 740 and low order part multiplexer760.

Redundant digits lookup table unit 730 and low order part multiplexer750 operate in the manner previously described for redundant digitslookup table unit 530 and low order part multiplexer 540. That is, firstredundant digits lookup table 730 receives the first nine (9) leadingbits (bit “b1” through bit “b9”) from input latch 710 and low order partmultiplexer 750 receives the tenth through twelfth bits (bit “b10”through “bit12”) from input latch 710. First redundant digits lookuptable 730 provides forty (40) bits to low order part multiplexer 750 aspreviously described. However, for reasons that will be explained,redundant digits lookup table 300 in redundant digits lookup table unit730 does not contain table entries having the leading three (3) bits setto a value of one (“1”). The output numeric value from first redundantdigits lookup table 730 and low order part multiplexer 750 is providedto multiplexer 770.

When read only memory (ROM) select unit 720 reads a bit combination ofthree (3) ores (“111”) then read only memory (ROM) select unit 720activates power in that portion of compressed direct lookup table unit700 that comprises second redundant digits lookup table unit 740 and loworder part multiplexer 760. Power is not activated in the portion ofcompressed direct lookup table unit 700 that comprises first redundantdigits lookup table unit 730 and low order part multiplexer 750.

Redundant digits lookup table unit 740 and low order part multiplexer760 operate in the manner previously described for redundant digitslookup table unit 530 and low order part multiplexer 540 except that (1)second redundant digits lookup table unit 740 receives the fourththrough ninth bits (bit “b4” through “bit9”) from input latch 710 and(2) redundant digits lookup table 300 in redundant digits lookup tableunit 740 contains only those table entries that have the leading three(3) bits set to a value of one (“1”).

Second redundant digits lookup table unit 740 is activated only forinput numeric values that have bits “b1” through “b3” set to a value ofone (“1”). This means that redundant digits lookup table unit 740 onlypasses twenty four (24) bits to low order part multiplexer 760(representing eight (8) sets of three (3) bits of the low order digitsparts). Redundant digits lookup table unit 740 passes eleven (11) bitsdirectly to multiplexer 770. Low order part multiplexer 760 passes tomultiplexer 770 three (3) output bits that represent the low order partof the output.

Read only memory (ROM) select unit 720 sends an enable signal tomultiplexer 770 to select (1) a first output numeric value from firstredundant digits lookup table unit 730 and low order part multiplexer750, or (2) a second output numeric value from second redundant digitslookup table unit 740 and low order part multiplexer 760. Multiplexer770 provides the selected output numeric value to result latch 780.

The alternate advantageous embodiment of the invention shown in FIG. 7is an example of a compressed direct lookup table unit that comprisesmore than one redundant digits table lookup unit. The present inventionis not limited to using only two redundant digits lookup table units.Three or more such units may be used. The alternate advantageousembodiment shown in FIG. 7 is but one example of a compressed directlookup table unit that uses a plurality of redundant digits lookup tableunits.

The use of two or more redundant digits lookup table units provides morecompression. In addition, because only one portion of the compresseddirect lookup table unit is powered up at any particular time, power isconserved in the operation of the unit.

If the size of reciprocal binary lookup table 300 is increased to a“fourteen (14) bits in, fourteen (14) bits out” table, then the outputwould partition 413 into seven “radix 4” digits. If the size ofreciprocal binary lookup table 300 is increased to a “sixteen (16) bitsin, sixteen (16) bits out” table, then the output would partition 5|3into eight “radix 4” digits. In other words, as the size of “k” grows,the high order output partition size also grows. However, the low orderpartition remains the same size on each line improving the compression.

Compression measured at the digit level is about fifty percent (50%) inreciprocal binary lookup table 300 because the high order fraction digittriple d₁d₂d₃ is stored only once per eight (8) low order digit tripled₄d₅d₆ (recall that the leading normalized digit d₀ does not need to bestored). For an “eighteen (18) bits in, eighteen (18) bits out” tablethe compression has improved down to almost thirty three percent (33%).The lookup procedure for a much larger table could be iterated to yielda somewhat greater compression.

For example, an “eighteen (18) bits in, eighteen (18) bits out” tablecould have the input partitioned into three parts of size 11|4|3. Insuch a case the eleven (11) leading fraction bits would index a block ofsixteen (16) lines. The next four (4) bits would index a line of theblock. The leading four (4) output “radix 4” fraction digits could belisted just once for the block with the next two (2) “radix 4” digitsfound on the line. The final three (3) “radix 4” digits are selectedfrom a column entry on the line. Here the first of the two middle digitsfor the sixteen (16) output lines of the block will have at most two (2)successive values, so that the redundant digit set ensures that nooverflow situation occurs. Concatenation of the leading four (4) outputfraction digits of the block, the next two (2) output digits from theindexed line of the block, and the final three (3) output digitsselected from a column entry on that line provides the nine (9) fractiondigit output.

It should be clear to one skilled in the art that many hierarchicalchoices to achieve greater compression can be achieved with theredundancy in the digit values allowing straightforward concatenation ofparts to compute the result. It should also be clear that redundantoutput in a higher radix is achievable in the same manner. A “twelve(12) bits in, twelve (12) bits out” table could have the input bitspartition 1012 with the output recoded to four (4) “radix 8” fractiondigits in the range [−4, 4] partitioned 2|2. Each of the 2¹⁰ outputlines would contain a leading fraction part of two (2) “radix 8” digitsfollowed by four (4) low order parts of two (2) “radix 8” digits each.The line corresponding to the first half of line seven from reciprocalbinary lookup table 300 would become:

TABLE 3 00 01 10 11 1.000 000 110 0: 2.0{overscore (2)} 40 4{overscore(2)} 4{overscore (4)} 42

FIG. 8 illustrates a flow chart of an advantageous embodiment of amethod of the present invention for providing a compressed direct lookuptable unit. The steps of the method shown in FIG. 8 are generallydenoted with reference numeral 800.

An input numeric value is represented by a plurality of input bits. Inone embodiment of the method of the invention a first portion of theinput bits are sent to redundant digits lookup table unit 530 (step810). A second portion of the input bits are sent to low order partmultiplexer 540 (step 820).

Redundant digits lookup table unit 530 accesses redundant digits lookuptable 300 to obtain a high order digits part of an output numeric valueand to obtain a plurality of low order digits part of the output numericvalue (step 830). Redundant digits lookup table unit 530 sends theplurality of low order digits parts to low order part multiplexer 540(step 840).

Low order part multiplexer 540 selects one of the low order digits partsfor the low order output numeric value from the plurality of low orderdigits parts (step 85D). The high order digit part from redundant digitslookup table unit 530 and the selected low order digits part from thelow order part multiplexer 540 are concatenated to form the outputnumeric value (step 860).

The above examples and description have been provided only for thepurpose of illustration, and are not intended to limit the invention inany way. As will be appreciated by the skilled person, the invention canbe carried out in a great variety of ways, employing more than onetechnique from those described above, all without exceeding the scope ofthe invention.

APPENDIX ONE LOGARITHMS (Continued) N. 0 1 2 3 4 5 6 7 8 9 d. 1150 0608975 7356 7734 8111 8489 8866 9244 9621 9999 *0376 378 1151 061 07531131 1508 1885 2262 2639 3017 3394 3771 4148 377 1152 4525 4902 52795656 6032 6409 6786 7163 7540 7916 377 1153 8293 8670 9046 9423 9799*0176 *0552 *0929 *1305 *1682 377 1154 062 2058 2434 2811 3187 3563 39394316 4692 5068 5444 376 1155 5820 6196 6572 6948 7324 7699 8075 84518827 9203 376 1156 9578 9954 *0330 *0706 *1081 *1456 *1832 *2207 *2583*2958 376 1157 063 3334 3709 4084 4460 4835 5210 5585 5960 6335 6711 3761158 7086 7461 7836 8211 8585 8960 9335 9710 *0085 *0460 376 1159 0640834 1209 1584 1958 2333 2708 3082 3457 3831 4205 376 1160 4580 49545329 5703 6077 6451 6826 7200 7574 7948 374 1161 8322 8696 9070 94449818 *0192 *0666 *0940 *1314 *1688 374 1162 065 2061 2435 2809 3182 35563930 4303 4677 6060 5424 374 1163 5797 6171 6544 6917 7291 7664 90378410 8784 9157 373 1164 9530 9903 *0276 *0649 *1022 *1395 *1768 *2141*2514 *2886 373 1165 066 3259 3832 4005 4377 4750 5123 5495 5868 62416613 373 1166 6085 7353 7730 8103 8475 8847 9220 9592 9964 *0336 3721167 067 0709 1081 1453 1825 2197 2589 2941 3313 3585 4057 372 1168 44284800 5172 5544 5915 6287 6659 7030 7402 7774 372 1169 8145 8517 88889259 9631 *0002 *0374 *0746 *1116 *1487 371 1170 068 1859 2230 2601 29723343 3714 4085 4456 4827 5198 371 1171 5569 5940 6311 6681 7052 74237794 8164 8535 8906 371 1172 9576 9647 *0017 *0388 *0758 *1129 *1499*1869 *2240 *2810 370 1173 069 2980 3350 3721 4091 4461 4831 5201 55715941 5311 370 1174 6681 7051 7421 7791 8160 8530 8900 9270 9639 *0009370 1175 070 0379 0748 1118 1487 1857 2226 2396 2965 3336 3704 369 11764073 4442 4812 5181 5550 5919 6288 6658 7027 7396 369 1177 7766 81348503 8871 9240 9609 9978 *0347 *0715 *1084 369 1178 071 1463 1822 21902559 2927 3296 3664 4033 4401 4770 369 1179 5138 5506 5875 6243 66116979 7348 7716 8084 8452 368 1180 8820 9188 9556 9924 *0292 *0650 *1028*1396 *1763 *2131 368 1181 072 2499 2867 3234 3602 3970 4337 4705 50726440 5807 368 1182 6175 6542 6910 7277 7644 8011 8379 8746 9113 9480 3671183 9647 *0215 *0582 *0949 *1316 *1683 *2050 *2416 *2783 *3150 367 1184073 3517 3884 4251 4617 4984 5351 5717 6084 6450 6817 367 1185 7184 75507916 8283 8649 9016 9382 9748 *0114 *0481 366 1186 074 0847 1213 15791945 2311 2677 3043 3409 3776 4141 366 1187 4507 4873 5239 5805 59708338 8702 7068 7433 7799 366 1188 8164 8530 3895 9261 9626 9992 *0357*0723 *1088 *1453 365 1189 075 1819 2184 2549 2914 3279 3844 4010 43754740 5106 365 1190 5470 5835 6199 6564 6929 7294 7659 8024 8388 8753 3651191 9118 9482 9847 *0211 *0576 *0940 *1305 *1669 *2034 *2398 364 1192076 2763 3127 3491 3855 4220 4584 4948 5312 5676 6040 364 1193 6404 67887132 7496 7860 8224 8588 8952 9316 9680 364 1194 077 0043 0407 0771 11341498 1862 2225 2589 2952 3316 364 1195 3679 4042 4406 4769 5133 54965859 6222 6585 6949 363 1196 7312 7675 8038 6401 8764 9127 9490 9853*0216 *0579 363 1197 078 0942 1304 1867 2030 2393 2755 3118 3480 38434206 363 1198 4568 4931 5293 6856 6018 6380 6743 7105 7487 7830 362 11998192 8554 8916 9278 9640 *0003 *0368 *0727 *1089 *1451 362 1200 079 18122174 2536 2898 3260 3622 3983 4345 4707 5068 362 N. 0 1 2 3 4 5 6 7 8 9d.

APPENDIX TWO 000 001 010 011 100 101 110 111 Part 11.000000000000(1)-1.000011111111(1) 1.000000000: +2.+0+0+0 +0+0−1 +0−1+1+0−1−1 +0−2+1 +0−2−1 −1+1+1 −1+1−1 −1+0+1 1.000000001: +2.+0+0+0 −1+0−1−1−1+1 −1−1−1 −1−2+1 −1−2−1 −2+1+1 −2+1−1 −2+0+1 1.000000010: +2.+0+0−1+2+0−1 +2−1+1 +2−1−1 +2−2+1 +2−2−1 +1+1+1 +1+1−1 +1+0+1 1.000000011:+2.+0+0−1 +1+0−1 +1−1+1 +1−1−1 +1−2+1 +1−2−1 +0+1+1 +0+1−1 +0+0+11.000000100: +2.+0+0−1 +0+0+0 +0+0−2 +0−1+0 +0−1−2 +0−2+0 +0−2−2 −1+1+0−1+1−2 1.000000101: +2.+0+0−1 −1+0+0 −1+0−2 −1−1+0 −1−1−2 −1−2+0 −1−2−2−2+1+0 −2+1−2 1.000000110: +2.+0+0−2 +2+0+0 +2+0−2 +2−1+0 +2−1−2 +2−2+0+2−2−2 +1+1+0 +1+1−2 1.000000111: +2.+0+0−2 +1+0+1 +1+0−1 +1−1+1 +1−1−1+1−2+1 +1−2−1 +0+1+1 +0+1−1 1.000001000: +2.+0+0−2 +0+0+1 +0+0−1 +0−1+1+0−1−1 +0−2+1 +0−2−1 −1+1+1 −1+1−1 1.000001001: +2.+0+0−2 −1−1−2 −1+0+0−1+0−2 −1−1+0 −1−1−2 −1−2+0 −1−2−2 −2+1+0 1.000001010: +2.+0−1+1 +2+1−2+2+0+0 +2+0−2 +2−1+0 +2−1−2 +2−2+0 +2−2−1 +1+1+1 1.000001011: +2.+0−1+1+1+1−1 +1+0+1 +1+0−1 +1−1+1 +1−1−1 +1−2+1 +1−2−1 +0+1+1 1.000001100:+2.+0−1+1 +0+1−1 +0+1−2 +0+0+0 +0+0−2 +0−1+0 +0−1−2 +0−2+0 +0−2−21.000001101: +2.+0−1+1 −1+1+0 −1+1−2 −1+0+0 −1+0−2 −1−1+1 −1−1−1 −1−2+1−1−2−1 1.000001110: +2.+0−1+1 −2+1+1 −2+1−1 −2+0+1 −2+0−1 −2−1+1 −2−1+0−2−1−2 −2−2+0 1.000001111: +2.+0−1+0 +2−2−2 +1+1+0 +1+1−2 +1+0+0 +1+0−2+1−1+0 +1−1−1 +1−2+1 1.000010000: +2.+0−1+0 +1−2−1 +0+1+1 +0+1−1 +0+0+1+0+0−1 +0−1+1 +0−1+0 +0−1−2 1.000010001: +2.+0−1+0 +0−2+0 +0−2−2 −1+1+0−1+1−2 −1+0+0 −1+0−2 −1−1+1 −1−1−1 1.000010010: +2.+0−1+0 −1−2+1 −1−2−1−2+1+1 −2+1−1 −2+0+1 −2+0+0 −2+0−2 −2−1+0 1.000010011: +2.+0−1−1 +2−1−2+2−2+0 +2−2−2 +1+1+0 +1+1−1 +1+0+1 +1+0−1 +1−1+1 1.000010100: +2.+0−1−1+1−1−1 +1−2+1 +1−2−1 +1−2−2 +0+1+0 +0+1−2 +0+0+0 +0+0−2 1.000010101:+2.+0−1−1 +0−1+0 +0−1−2 +0−2+1 +0−2−1 −1+1+1 −1+1−1 −1+0+1 −1+0−11.000010110: +2.+0−1−1 −1+0−2 −1−1+0 −1−1−2 −1−2+0 −1−2−2 −2+1+0 −2+1−1−2+0+1 1.000010111: +2.+0−1−2 +2+0−1 +2−1+1 +2−1−1 +2−2+1 +2−2+0 +2−2−2+1+1+0 +2+1−2 1.000011000: +2.+0−1−2 +1+0+0 +1+0−2 +1−1+1 +1−1−1 +1−2+1+1−2−1 +0+1+1 +0+1+0 1.000011001: +2.+0−1−2 +0+1−2 +0+0+0 +0+0−2 +0−1+0+0−1−2 +0−2+1 +0−2−1 −1+1+1 1.000011010: +2.+0−1−2 −1+1−1 −1+0+1 −1+0+0−1+0−2 −1−1+0 −1−1−2 −1−2+0 −1−2−1 1.000011011: +2.+0−1−2 −2+1+1 −2+1−1−2+0+1 −2+0−1 −2+0−2 −2−1+0 −2−1−2 −2−2+0 1.000011100: +2.+0−2+1 +2−2−2+1+1+1 +1+1−1 +1+0+1 +1+0−1 +1−1+1 +1−1+0 +1−1−2 1.000011101: +2.+0−2+1+1−2+0 +1−2−2 +0+1+0 +0+1−1 +0+0+1 +0+0−1 +0−1+1 +0−1−1 1.000011110:+2.+0−2+1 +0−1−2 +0−2+0 +0−2−2 −1+1+0 −1+1−1 −1+0+1 −1+0−1 −1−1+11.000011111: +2.+0−2+1 −1−1−1 −1−1−2 −1−2+0 −1−2−2 −2+1+0 −2+1−1 −2+0+1−2+0−1 Part 2 1.000100000000(1)-1.000111111111(1) 1.000100000: +2.+0−2+0+2−1+1 +2−1−1 +2−1−2 +2−2+0 +2−2−2 +1+1+0 +1+1−1 +1+0+1 1.000100001:+2.+0−2+0 +1+0−1 +1−1+1 +1−1+0 +1−1−2 +1−2+0 +1−2−2 +0+1+1 +0+1−11.000100010: +2.+0−2+0 +0+0+1 +0+0−1 +0−1+1 +0−1+0 +0−1−2 +0−2+0 +0−2−2−1+1+1 1.000100011: +2.+0−2+0 −1+1−1 −1+0+1 −1+0−1 −1+0−2 −1−1+0 −1−1−2−1−2+0 −1−2−1 1.000100100: +2.+0−2+0 −2+1+1 −2+1−1 −2+0+1 −2+0+0 −2+0−2−2−1+0 −2−1−1 −2−2+1 1.000100101: +2.+0−2−1 +2−2−1 +1+1+1 +1+1+0 +1+1−2+1+0+0 +1+0−2 +1−1+1 +1−1−1 1.000100110: +2.+0−2−1 +1−2+1 +1−2−1 +1−2−2+0+1+0 +0+1−2 +0+0+0 +0+0−1 +0−1+1 1.000100111: +2.+0−2−1 +0−1−1 +0−1−2+0−2+0 +0−2−2 −1+1+0 −1+1−1 −1+0+1 −1+0−1 1.000101000: +2.+0−2−1 −1+0−2−1−1+0 −1−1−2 −1−2+0 −1−2−1 −2+1+1 −2+1−1 −2+0+1 1.000101001: +2.+0−2−2+2+0+0 +2+0−2 +2−1+0 +2−1−1 +2−2+1 +2−2−1 +2−2−2 +1+1+0 1.000101010:+2.+0−2−2 +1+1−2 +1+0+0 +1+0−1 +1−1+1 +1−1−1 +1−1−2 +1−2+0 +1−2−21.000101011: +2.+0−2−2 +0+1+0 +0+1−1 +0+0+1 +0+0−1 +0+0−2 +0−1+0 +0−1−2+0−2+1 1.000101100: +2.+0−2−2 +0−2−1 −1+1+1 −1+1−1 −1+1−2 −1+0+0 −1+0−2−1−1+1 −1−1−1 1.000101101: +2.+0−2−2 −1−2+1 −1−2+0 −1−2−2 −2+1+0 −2+1−1−2+0+1 −2+0−1 −2+0−2 1.000101110: +2.−1+1+1 +2−1+0 +2−1−2 +2−2+0 +2−2−1+1+1+1 +1+1−1 +1+1−2 +1+0+0 1.000101111: +2.−1+1+1 +1+0−2 +1−1+1 +1−1−1+1−2+1 +1−2+0 +1−2−2 +0+1+0 +0+1−1 1.000110000: +2.−1+1+1 +0+0+1 +0+0−1+0+0−2 +0−1+0 +0−1−2 +0−2+1 +0−2−1 −1+1+1 1.000110001: +2.−1+1+1 −1+1+0−1+1−2 −1+0+0 −1+0−1 −1−1+1 −1−1−1 −1−1−2 −1−2+0 1.000110010: +2.−1+1+1−1−2−2 −2+1+1 −2+1−1 −2+0+1 −2+0+0 −2+0−2 −2−1+0 −2−1−1 1.000110011:+2.−1+1+0 +2−2+1 +2−2−1 +2−2−2 +1+1+0 +1+1−2 +1+0+1 +1+0−1 +1+0−21.000110100: +2.−1+1+0 +1−1+0 +1−1−2 +1−2+1 +1−2−1 +0+1+1 +0+1+0 +0+1−2+0+0+0 1.000110101: +2.−1+1+0 +0+0−1 +0−1+1 +0−1−1 +0−1−2 +0−2+0 +0−2−1−1+1+1 −1+1−1 1.000110110: +2.−1+1+6 −1+1−2 −1+0+0 −1+0−2 −1−1+1 −1−1−1−1−2+1 −1−2+0 −1−2−2 1.000110111: +2.−1+1+0 −2+1+1 −2+1−1 −2+0+1 −2+0+0−2+0−2 −2−1+0 −2−1−1 −2−2+1 1.000111000: +2.−1+1−1 +2−2+0 +2−2−2 +1+1+0+1+1−1 +1+0+1 +1+0−1 +1+0−2 +1−1+0 1 000111001: +2.−1+1−1 +1−1−1 +1−2+1+1−2−1 +1−2−2 +0+1+0 +0+1−2 +0+0+1 +0+0−1 1.000111010: +2.−1+1−1 +0+0−2+0−1+0 +0−1−2 +0−2+1 +0−2−1 +0−2−2 −1+1+0 −1+1−2 1.000111011: +2.−1+1−1−1+0+1 −1+0−1 −1+0−2 −1−1+0 −1−1−2 −1−2+1 −1−2−1 −1−2−2 1.000111100:+2.−1+1−1 −2+1+0 −2+1−2 −2+0+1 −2+0−1 −2−1+1 −2−1+0 −2−1−2 −2−2+11.000111101: +2.−1+1−2 +2−2−1 +2−2−2 +1+1+0 +1+1−2 +1+0+1 +1+0−1 +1+0−2+1−1+0 1.000111110: +2.−1+1−2 +1−1−2 +1−2+1 +1−2−1 +1−2−2 +0+1+0 +0+1−2+0+0+1 +0+0−1 1.000111111: +2.−1+1−2 +0+0−2 +0−1+0 +0−1−2 +0−2+1 +0−2−1+0−2−2 −1+1+0 −1+1−1 Part 3 1.001000000000(1)-1.001011111111(1)1.001000000: +2.−1+1−2 −1+0+1 −1+0−1 −1+0−2 −1−1+0 −1−1−1 −1−2+1 −1−2+0−1−2−2 1.001000001: +2.−1+1−2 −2+1+0 −2+1−1 −2+0+1 −2+0+0 −2+0−2 −2−1+1−2−1−1 −2−2+1 1.001000010: +2.−1+0+1 +2−2+0 +2−2−2 +1+1+1 +1+1−1 +1+1−2+1+0+0 +1+0−2 +1−1+1 1.001000011: +2.−1+0+1 +1−1−1 +1−1−2 +1−2+0 +1−2−1+0+1+1 +0+1−1 +0+1−2 +0+0+0 1.001000100: +2.−1+0+1 +0+0−1 +0−1+1 +0−1+0+0−1−2 +0−2+1 +0−2−1 −1+1+1 −1+1+0 1.001000101: +2.−1+0+1 −1+1−2 −1+0+1−1+0−1 −1+0−2 −1−1+0 −1−1−1 −1−2+1 −1−2−1 1.001000110: +2.−1+0+1 −1−2−2−2+1+0 −2+1−1 −2+0+1 −2+0+0 −2+0−2 −2−1+1 −2−1−1 1.001000111: +2.−1+0+0+2−1−2 +2−2+0 +2−2−2 +1+1+1 +1+1−1 +1+1−2 +1+0+0 +1+0−1 1.001001000:+2.−1+0+0 +1−1+1 +1−1+0 +1−1−2 +1−2+1 +1−2−1 +1−2−2 +0+1+0 +0+1−11.001001001: +2.−1+0+0 +0+0+1 +0+0−1 +0+0−2 +0−1+0 +0−1−1 +0−2+1 +0−2+0+0−2−2 1.001001010: +2.−1+0+0 −1+1+1 −1+1−1 −1+1−2 −1+0+0 −1+0−1 −1−1+1−1−1+0 −1−1−2 1.001001011: +2.−1+0+0 −1−2+1 −1−2−1 −1−2−2 −2+1+0 −2+1−2−2−0+1 −2+0−1 −2+0−2 1.001001100: +2.−1+0−1 +2−1+0 +2−1−1 +2−2+1 +2−2+0+2−2−2 +1+1+1 +1+1−1 +1+1−2 1.001001101: +2.−1+0−1 +1+0+0 +1+0−1 +1−1+1+1−1+0 +1−1−2 +1−2+1 +1−2−1 +1−2−2 1.001001110: +2.−1+0−1 +0+1+0 +0+1−1+0+0+1 +0+0+0 +0+0−2 +0−1+1 +0−1−1 +0−1−2 1.001001111: +2.−1+0−1 +0−2+0+0−2−1 −1+1+1 −1+1+0 −1+1−2 −1+0+1 −1+0−1 −1+0−2 1.001010000: +2.−1+0−1−1−1+0 −1−1−1 −1−2+1 −1−2+0 −1−2−2 −2+1+1 −2+1−1 −2+1−2 1.001010001:+2.−1+0−1 −2+0+0 −2+0−1 −2−1+1 −2−1+0 −2−1−2 −2−2+1 −2−2−1 −2−2−21.001010010: +2.−1+0−2 +1+1+0 +1+1−1 +1+0+1 +1+0+0 +1+0−2 +1−1+1 +1−1−1+1−1−2 1.001010011: +2.−1+0−2 +1−2+1 +1−2−1 +1−2−2 +0+1+0 +0+1−1 +0+0+1+0+0+0 +0+0−2 1.001010100: +2.−1+0−2 +0−1+1 +0−1−1 +0−1−2 +0−2+0 +0−2−1−1+1+1 −1+1+0 −1+1−2 1.001010101: +2.−1+0−2 −1+0+1 −1+0−1 −1+0−2 −1−1+0−1−1−1 −1−1−2 −1−2+0 −1−2−1 1.001010110: +2.−1+0−2 −2+1+1 −2+1+0 −2+1−2−2+0+1 −2+0−1 −2+0−2 −2−1+0 −2−1−1 1.001010111: +2.−1−1+1 +2−2+1 +2−2+0+2−2−1 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+1−2+0 +1−2−21.001100100: +2.−1−1−1 +0+1+1 +0+1−1 +0+1−2 +0+0+1 +0+0−1 +0+0−2 +0−1+0+0−1−1 1.001100101: +2.−1−1−1 +0−1−2 +0−2+0 +0−2−1 −1+1+1 −1+1+0 −1+1−1−1+0+1 −1+0+0 1.001100110: +2.−1−1−1 −1+0−2 −1−1+1 −1−1+0 −1−1−2 −1−2+1−1−2−1 −1−2−2 −2+1+1 1.001100111: +2.−1−1−1 −2+1−1 −2+1−2 −2+0+1 −2+0−1−2+0−2 −2−1+0 −2−1−1 −2−1−2 1.001101000: +2.−1−1−2 +2−2+0 +2−2−1 +1+1+1+1+1+0 +1+1−1 +1+0+1 +1+0+0 +1+0−1 1.001101001: +2.−1−1−2 +1−1+1 +1−1+0+1−1−2 +1−2+1 +1−2+0 +1−2−2 +0+1+1 +0+1+0 1.001101010: +2.−1−1−2 +0+1−2+0+0+1 +0+0−1 +0+0−2 +0−1+1 +0−1−1 +0−1−2 +0−2+1 1.001101011: +2.−1−1−2+0−2−1 +0−2−2 −1+1+1 −1+1−1 −1+1−2 −1+0+0 −1+0−1 −1+0−2 1.001101100:+2.−1−1−2 −1−1+0 −1−1−1 −1−1−2 −1−2+0 −1−2−1 −1−2−2 −2+1+0 −2+1−11.001101101: +2.−1−1−2 −2+0+1 −2+0+0 −2+0−1 −2−1+1 −2−1+0 −2−1−1 −2−2+1−2−2+0 1.001101110: +2.−1−2+1 +2−2−1 +1+1+1 +1+1+0 +1+1−1 +1+0+1 +1+0+0+1+0−2 +1−1+1 1.001101111: +2.−1−2+1 +1−1+0 +1−1−2 +1−2+1 +1−2+0 +1−2−2+0+1+1 +0+1+0 +0+1−2 1.001110000: +2.−1−2+1 +0+0+1 +0+0+0 +0+0−2 +0−1+1+0−1+0 +0−1−2 +0−2+1 +0−2+0 1.001110001: +2.−1−2+1 +0−2−2 −1+1+1 −1+1+0−1+1−2 −1+0+1 −1+0+0 −1+0−2 −1−1+1 1.001110010: +2.−1−2+1 −1−1−1 −1−1−2−1−2+1 −1−2−1 −1−2−2 −2+1+1 −2+1−1 −2+1−2 1.001110011: +2.−1−2+1 −2+0+1−2+0−1 −2+0−2 −2−1+1 −2−1−1 −2−1−2 −2−2+1 −2−2−1 1.001110100: +2.−1−2+0+2−2−2 +1+1+1 +1+1+0 +1+1−2 +1+0+1 +1+0+0 +1+0−2 +1−1+1 1.001110101:+2.−1−2+0 +1−1+0 +1−1−2 +1−2+1 +1−2+0 +1−2−2 +0+1+1 +0+1+0 +0+1−21.001110110: +2.−1−2+0 +0+0+1 +0+0+0 +0+0−2 +0−1+1 +0−1+0 +0−1−2 +0−2+1+0−2+0 1.001110111: +2.−1−2+0 +0−2−2 −1+1+1 −1+1+0 −1+1−2 −1+0+1 −1+0+0−1+0−1 −1−1+1 1.001111000: +2.−1−2+0 −1−1+0 −1−1−1 −1−2+1 −1−2+0 −1−2−1−2+1+1 −2+1+0 −2+1−1 1.001111001: +2.−1−2+0 −2+0+1 −2+0+0 −2+0−1 −2−1+1−2−1+0 −2−1−1 −2−1−2 −2−2+0 1.001111010: +2.−1−2−1 +2−2−1 +2−2−2 +1+1+0+1+1−1 +1+1−2 +1+0+0 +1+0−1 +1+0−2 1.001111011: +2.−1−2−1 +1−1+1 +1−1−1+1−1−2 +1−2+1 +1−2−1 +1−2−2 +0+1+1 +0+1−1 1.001111100: +2.−1−2−1 +0+1−2+0+0+1 +0+0+0 +0+0−2 +0−1+1 +0−1+0 +0−1−2 +0−2+1 1.001111101: +2.−1−2−1+0−2+0 +0−2−1 −1+1+1 −1+1+0 −1+1−1 −1+0+1 −1+0+0 −1+0−1 1.001111110:+2.−1−2−1 −1+0−2 −1−1+0 −1−1−1 −1−1−2 −1−2+0 −1−2−1 −1−2−2 −2+1+01.001111111: +2.−1−2−1 −2+1−1 −2+1−2 −2+0+1 −2+0−1 −2+0−2 −2−1+1 −2−1+0−2−1−2 Part 5 1.010000000000(1)-1.010011111111(1) 1.010000000: +2.−1−2−2+2−2+1 +2−2+0 +2−2−2 +1+1+1 +1+1+0 +1+1−1 +1+0+1 +1+0+0 1.010000001:+2.−1−2−2 +1+0−1 +1−1+1 +1−1+0 +1−1−1 +1−1−2 +1−2+0 +1−2−1 +1−2−21.010000010: +2.−1−2−2 +0+1+1 +0+1−1 +0+1−2 +0+0+1 +0+0−1 +0+0−2 +0−1+1+0−1+0 1.010000011: +2.−1−2−2 +0−1−2 +0−2+1 +0−2+0 +0−2−1 −1+1+1 −1+1+0−1+1−1 −1+1−2 1.010000100: +2.−1−2−2 −1+0+0 −1+0−1 −1+0−2 −1−1+0 −1−1−1−1−1−2 −1−2+1 −1−2−1 1.010000101: +2.−1−2−2 −1−2−2 −2+1+1 −2+1+0 −2+1−2−2+0+1 −2+0+0 −2+0−1 −2−1+1 1.010000110: +2.−2+1+1 +2−1+0 +2−1−1 +2−1−2+2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1−1 1.010000111: +2.−2+1+1 +1+1−2 +1+0+1+1+0+0 +1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−2+1 1.010001000: +2.−2+1+1 +1−2+0+1−2−1 +1−2−2 +0+1+0 +0+1−1 +0+1−2 +0+0+1 +0+0−1 1.010001001: +2.−2+1+1+0+0−2 +0−1+1 +0−1+0 +0−1−2 +0−2+1 +0−2+0 +0−2−1 −1+1+1 1.010001010:+2.−2+1+1 −1+1+0 −1+1−1 −1+1−2 −1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1−11.010001011: +2.−2+1+1 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −2+1+1 −2+1+0 −2+1−1−2+1−2 1.010001100: +2.−2+1+1 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1−1 −2−1−2−2−2+1 −2−2+0 1.010001101: +2.−2+1+0 +2−2−1 +1+1+1 +1+1+0 +1+1−1 +1+1−2+1+0+0 +1+0−1 +1+0−2 1.010001110: +2.−2+1+0 +1−1+1 +1−1−1 +1−1−2 +1−2+1+1−2+0 +1−2−1 +0+1+1 +0+1+0 1.010001111: +2.−2+1+0 +0+1−1 +0+1−2 +0+0+0+0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−2 1.010010000: +2.−2+1+0 +0−2+1 +0−2+0+0−2−1 −1+1+1 −1+1+0 −1+1−1 −1+1−2 −1+0+1 1.010010001: +2.−2+1+0 −1+0−1−1+0−2 −1−1+1 −1−1+0 −1−1−1 −1−2+1 −1−2+0 −1−2−1 1.010010010: +2.−2+1+0−1−2−2 −2+1+1 −2+1−1 −2+1−2 −2+0+1 −2+0+0 −2+0−2 −2−1+1 1.010010011:+2.−2+1−1 +2−1+0 +2−1−1 +2−1−2 +2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+01.010010100: +2.−2+1−1 +1+1−2 +1+0+1 +1+0+0 +1+0−1 +1+0−2 +1−1+0 +1−1−1+1−1−2 1.010010101: +2.−2+1−1 +1−2+1 +1−2+0 +1−2−2 +0+1+1 +0+1+0 +0+1−1+0+1−2 +0+0+0 1.010010110: +2.−2+1−1 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−2+0−2+1 +0−2+0 +0−2−1 1.010010111: +2.−2+1−1 +0−2−2 −1+1+0 −1+1−1 −1+1−2−1+0+1 −1+0+0 −1+0−1 −1−1+1 1.010011000: +2.−2+1−1 −1−1+0 −1−1−1 −1−1−2−1−2+1 −1−2−1 −1−2−2 −2+1+1 −2+1+0 1.010011001: +2.−2+1−1 −2+1−1 −2+0+1−2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−2 1.010011010: +2.−2+1−2 +2−2+1+2−2+0 +2−2−1 +2−2−2 +1+1+0 +1+1−1 +1+1−2 +1+0+1 1.010011011: +2.−2+1−2+1+0+0 +1+0−1 +1−1+1 +1−1+0 +1−1−1 +1−1−2 +1−2+1 +1−2−1 1.010011100:+2.−2+1−2 +1−2−2 +0+1+1 +0+1+0 +0+1−1 +0+1−2 +0+0+0 +0+0−1 +0+0−21.010011101: +2.−2+1−2 +0−1+1 +0−1+0 +0−1−1 +0−2+1 +0−2+0 +0−2−1 +0−2−2−1+1+1 1.010011110: +2.−2+1−2 −1+1+0 −1+1−2 −1+0+1 −1+0+0 −1+0−1 −1+0−2−1−1+1 −1−1−1 1.010011111: +2.−2+1−2 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2−2+1+0 −2+1−1 −2+1−2 Part 6 1.010100000000(1)-1.010111111111(1)1.010100000: +2.−2+1−2 −2+0+1 −2+0+0 −2+0−1 −2−1+1 −2−1+0 −2−1−1 −2−1−2−2−2+1 1.010100001: +2.−2+0+1 +2−2+0 +2−2−1 +1+1+1 +1+1+0 +1+1−1 +1+1−2+1+0+1 +1+0+0 1.010100010: +2.−2+0+1 +1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−2+1−2+1 +1−2+0 +1−2−2 1.010100011: +2.−2+0+1 +0+1+1 +0+1+0 +0+1−1 +0+1−2+0+0+1 +0+0−1 +0+0−2 +0−1+1 1.010100100: +2.−2+0+1 +0−1+0 +0−1−1 +0−1−2+0−2+1 +0−2−1 +0−2−2 −1+1+1 −1+1+0 1.010100101: +2.−2+0+1 −1+1−1 −1+1−2−1+0+1 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 1.010100110: +2.−2+0+1 −1−1−2−1−2+1 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−2 1.010100111: +2.−2+0+1−2+0+1 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 −2−1−2 −2−2+1 1.010101000:+2.−2+0+0 +2−2+0 +2−2−2 +1+1+1 +1+1+0 +1+1−1 +1+1−2 +1+0+1 +1+0+01.010101001: +2.−2+0+0 +1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−2 +1−2+1 +1−2+0+1−2−1 1.010101010: +2.−2+0+0 +0+1+1 +0+1+0 +0+1−1 +0+1−2 +0+0+1 +0+0+0+0+0−1 +0+0−2 1.010101011: +2.−2+0+0 +0−1+0 +0−1−1 +0−1−2 +0−2+1 +0−2+0+0−2−1 +0−2−2 −1+1+1 1.010101100: +2.−2+0+0 −1+1−1 −1+1−2 −1+0+1 −1+0+0−1+0−1 −1+0−2 −1−1+1 −1−1+0 1.010101101: +2.−2+0+0 −1−1−1 −1−2+1 −1−2+0−1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 1.010101110: +2.−2+0+0 −2+1−2 −2+0+0−2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 −2−1−2 1.010101111: +2.−2+0−1 +2−2+1+2−2+0 +2−2−2 +1+1+1 +1+1+0 +1+1−1 +1+1−2 +1+0+1 1.010110000: +2.−2+0−1+1+0+0 +1+0−1 +1+0−2 +1−1+0 +1−1−1 +1−1−2 +1−2+1 +1−2+0 1.010110001:+2.−2+0−1 +1−2−1 +1−2−2 +0+1+1 +0+1+0 +0+1−1 +0+0+1 +0+0+0 +0+0−11.010110010: +2.−2+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 +0−1−2 +0−2+1 +0−2+0+0−2−2 1.010110011: +2.−2+0−1 −1+1+1 −1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+0−1+0−1 −1+0−2 1.010110100: +2.−2+0−1 −1−1+1 −1−1−1 −1−1−2 −1−2+1 −1−2+0−1−2−1 −1−2−2 −2+1+1 1.010110101: +2.−2+0−1 −2+1+0 −2+1−1 −2+1−2 −2+0+1−2+0−1 −2+0−2 −2−1+1 −2−1+0 1.010110110: +2.−2+0−2 +2−1−1 +2−1−2 +2−2+1+2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+0 1.010110111: +2.−2+0−2 +1+1−2 +1+0+1+1+0+0 +1+0−1 +1+0−2 +1−1+1 +1−1+0 +1−1−1 1.010111000: +2.−2+0−2 +1−1−2+1−2+1 +1−2+0 +1−2−1 +0+1+1 +0+1+0 +0+1−1 +0+1−2 1.010111001: +2.−2+0−2+0+0+1 +0+0+0 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 +0−1−2 1.010111010:+2.−2+0−2 +0−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+0 −1+1−1 −1+1−2 −1+0+11.010111011: +2.−2+0−2 −1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 −1−2+1−1−2+0 1.010111100: +2.−2+0−2 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−2−2+0+1 −2+0+0 1.010111101: +2.−2+0−2 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1−2−2+1 −2−2+0 −2−2−1 1.010111110: +2.−2−1+1 +2−2−2 +1+1+1 +1+1+0 +1+1−1+1+1−2 +1+0+1 +1+0+0 +1+0−1 1.010111111: +2.−2−1+1 +1+0−2 +1−1+1 +1−1+0+1−1−1 +1−1−2 +1−2+0 +1−2−1 +1−2−2 Part 71.011000000000(1)-1.011011111111(1) 1.011000000: +2.−2−1+1 +0+1+1 +0+1+0+0+1−1 +0+1−2 +0+0+1 +0+0+0 +0+0−1 +0+0−2 1.011000001: +2.−2−1+1 +0−1+1+0−1+0 +0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2−1 −1+1+1 1.011000010: +2.−2−1+1−1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1 1.011000011:+2.−2−1+1 −1−1+0 −1−1−1 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2 −2+1+11.011000100: +2.−2−1+1 −2+1+0 −2+1−1 −2+1−2 −2+0+0 −2+0−1 −2+0−2 −2−1+1−2−1+0 1.011000101: +2.−2−1+0 +2−1−1 +2−1−2 +2−2+1 +2−2+0 +2−2−1 +2−2−2+1+1+1 +1+1+0 1.011000110: +2.−2−1+0 +1+1−1 +1+1−2 +1+0+1 +1+0+0 +1+0−1+1+0−2 +1−1+1 +1−1+0 1.011000111: +2.−2−1+0 +1−1−1 +1−1−2 +1−2+1 +1−2+0+1−2−2 +0+1+1 +0+1+0 +0+1−1 1.011001000: +2.−2−1+0 +0+1−2 +0+0+1 +0+0+0+0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 1.011001001: +2.−2−1+0 +0−1−2 +0−2+1+0−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+0 −1+1−1 1.011001010: +2.−2−1+0 −1+1−2−1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 1.011001011: +2.−2−1+0−1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−2 1.011001100:+2.−2−1+0 −2+0+1 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 −2−1−21.011001101: +2.−2−1−1 +2−2+1 +2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+0 +1+1−1+1+1−2 1.011001110: +2.−2−1−1 +1+0+1 +1+0+0 +1+0−1 +1+0−2 +1−1+1 +1−1+0+1−1−1 +1−1−2 1.011001111: +2.−2−1−1 +1−2+1 +1−2+0 +1−2−1 +1−2−2 +0+1+1+0+1+0 +0+1−1 +0+1−2 1.011010000: +2.−2−1−1 +0+0+1 +0+0+0 +0+0−1 +0+0−2+0−1+1 +0−1+0 +0−1−1 +0−1−2 1.011010001: +2.−2−1−1 +0−2+1 +0−2+0 +0−2−1+0−2−2 −1+1+1 −1+1+0 −1+1−1 −1+1−2 1.011010010: +2.−2−1−1 −1+0+1 −1+0+0−1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 −1−1−2 1.011010011: +2.−2−1−1 −1−2+1−1−2+0 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−2 1.011010100: +2.−2−1−1−2+0+1 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 −2−1−2 1.011010101:+2.−2−1−2 +2−2+1 +2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+0 +1+1−1 +1+1−21.011010110: +2.−2−1−2 +1+0+1 +1+0+0 +1+0−1 +1+0−2 +1−1+1 +1−1+0 +1−1−1+1−1−2 1.011010111: +2.−2−1−2 +1−2+1 +1−2+0 +1−2−1 +1−2−2 +0+1+1 +0+1+0+0+1−1 +0+1−2 1.011011000: +2.−2−1−2 +0+0+1 +0+0+0 +0+0−1 +0+0−2 +0−1+1+0−1+0 +0−1−1 +0−1−2 1.011011001: +2.−2−1−2 +0−2+1 +0−2+0 +0−2−1 +0−2−2−1+1+1 −1+1+0 −1+1−1 −1+1−2 1.011011010: +2.−2−1−2 −1+0+1 −1+0+0 −1+0−1−1+0−2 −1−1+1 −1−1+0 −1−1−1 −1−1−2 1.011011011: +2.−2−1−2 −1−2+1 −1−2+0−1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−2 1.011011100: +2.−2−1−2 −2+0+1−2+0+0 −2+0−1 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 1.011011101: +2.−2−2+1+2−1−2 +2−2+1 +2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+0 +1+1−1 1.011011110:+2.−2−2+1 +1+1−2 +1+0+1 +1+0+0 +1+0−1 +1+0−2 +1−1+1 +1−1+0 +1−1−11.011011111: +2.−2−2+1 +1−1−2 +1−2+1 +1−2+0 +1−2−1 +1−2−2 +0+1+1 +0+1+0+0+1−1 Part 8 1.011100000000(1)-1.011111111111(1) 1.011100000: +2.−2−2+1+0+1−2 +0+0+1 +0+0+0 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1+0 1.011100001:+2.−2−2+1 +0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+01.011100010: +2.−2−2+1 −1+1−1 −1+1−2 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1−1−1+0 1.011100011: +2.−2−2+1 −1−1−1 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2−2+1+1 −2+1+0 1.011100100: +2.−2−2+1 −2+1−1 −2+1−1 −2+1−2 −2+0+1 −2+0+0−2+0−1 −2+0−2 −2−1+1 1.011100101: +2.−2−2+0 +2−1+0 +2−1−1 +2−1−2 +2−2+1+2−2+0 +2−2−1 +2−2−2 +1+1+1 1.011100110: +2.−2−2+0 +1+1+0 +1+1−1 +1+1−2+1+0+1 +1+0+0 +1+0−1 +1+0−1 +1+0−2 1.011100111: +2.−2−2+0 +1−1+1 +1−1+0+1−1−1 +1−1−2 +1−2+1 +1−2+0 +1−2−1 +1−2−2 1.011101000: +2.−2−2+0 +0+1+1+0+1+0 +0+1−1 +0+1−2 +0+0+1 +0+0+0 +0+0−1 +0+0−2 1.011101001: +2.−2−2+0+0−1+1 +0−1+1 +0−1+0 +0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2−1 1.011101010:+2.−2−2+0 +0−2−2 −1+1+1 −1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+0 −1+0−11.011101011: +2.−2−2+0 −1+0−2 −1−1+1 −1−1+1 −1−1+0 −1−1−1 −1−1−2 −1−2+1−1−2+0 1.011101100: +2.−2−2+0 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−2−2+0+1 −2+0+0 1.011101101: +2.−2−2+0 −2+0−1 −2+0−2 −2+0−2 −2−1+1 −2−1+0−2−1−1 −2−1−2 −2−2+1 1.011101110: +2.−2−2−1 +2−2+0 +2−2−1 +2−2−2 +1+1+1+1+1+0 +1+1−1 +1+1−2 +1+0+1 1.011101111: +2.−2−2−1 +1+0+0 +1+0+0 +1+0−1+1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−2 1.011110000: +2.−2−2−1 +1−2+1 +1−2+0+1−2−1 +1−2−2 +0+1+1 +0+1+0 +0+1+0 +0+1−1 1.011110001: +2.−2−2−1 +0+1−2+0+0+1 +0+0+0 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 1.011110010: +2.−2−2−1+0−1−2 +0−2+1 +0−2+0 +0−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+0 1.011110011:+2.−2−2−1 −1+1−1 −1+1−2 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1+01.011110100: +2.−2−2−1 −1−1+0 −1−1−1 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2−2+1+1 1.011110101: +2.−2−2−1 −2+1+0 −2+1−1 −2+1−2 −2+0+1 −2+0+1 −2+0+0−2+0−1 −2+0−2 1.011110110: +2.−2−2−1 −2−1+1 −2−1+0 −2−1−1 −2−1−2 −2−2+1−2−2+0 −2−2−1 −2−2−1 1.011110111: +2.−2−2−2 +2−2−2 +1+1+1 +1+1+0 +1+1−1+1+1−2 +1+0+1 +1+0+0 +1+0−1 1.011111000: +2.−2−2−2 +1+0−2 +1−1+1 +1−1+1+1−1+0 +1−1−1 +1−1−2 +1−2+1 +1−2+0 1.011111001: +2.−2−2−2 +1−2−1 +1−2−2+0+1+1 +0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1 1.011111010: +2.−2−2−2 +0+0+0+0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 +0−1−2 +0−1−2 1.011111011: +2.−2−2−2+0−2+1 +0−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+0 −1+1−1 −1+1−2 1.011111100:+2.−2−2−2 −1+0+1 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−11.011111101: +2.−2−2−2 −1−1−2 −1−2+1 −1−2+1 −1−2+0 −1−2−1 −1−2−2 −2+1+1−2+1+0 1.011111110: +2.−2−2−2 −2+1−1 −2+1−2 −2+0+1 −2+0+0 −2+0+0 −2+0−1−2+0−2 −2−1+1 1.011111111: +2.−2−2−2 −2−1+0 −2−1−1 −2−1−2 −2−2+1 −2−2+0−2−2+0 −2−2−1 −2−2−2 Part 9 1.100000000000(1)-1.100011111111(1)1.100000000: +1.+1+1+1 +1+1+1 +1+1+0 +1+1−1 +1+1−2 +1+0+1 +1+0+0 +1+0+0+1+0−1 1.100000001: +1.+1+1+1 +1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−2 +1−2+1+1−2+0 +1−2+0 1.100000010: +1.+1+1+1 +1−2−1 +1−2−2 +0+1+1 +0+1+0 +0+1−1+0+1−2 +0+0+1 +0+0+1 1.100000011: +1.+1+1+1 +0+0+0 +0+0−1 +0+0−2 +0−1+1+0−1+0 +0−1−1 +0−1−2 +0−2+1 1.100000100: +1.+1+1+1 +0−2+1 +0−2+0 +0−2−1+0−2−2 −1+1+1 −1+1+0 −1+1−1 −1+1−2 1.100000101: +1.+1+1+1 −1+1−2 −1+0+1−1+0+0 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 1.100000110: +1.+1+1+1 −1−1−1−1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2 −2+1+1 −2+1+0 1.100000111: +1.+1+1+1−2+1+0 −2+1−1 −2+1−2 −2+0+1 −2+0+0 −2+0−1 −2+0−2 −2−1+1 1.100001000:+1.+1+1+1 −2−1+1 −2−1+0 −2−1−1 −2−1−2 −2−2+1 −2−2+0 −2−2−1 −2−2−11.100001001: +1.+1+1+0 +2−2−2 +1+1+1 +1+1+0 +1+1−1 +1+1−2 +1+0+1 +1+0+0+1+0+0 1.100001010: +1.+1+1+0 +1+0−1 +1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−2+1−1−2 +1−2+1 1.100001011: +1.+1+1+0 +1−2+0 +1−2−1 +1−2−2 +0+1+1 +0+1+0+0+1−1 +0+1−1 +0+1−2 1.100001100: +1.+1+1+0 +0+0+1 +0+0+0 +0+0−1 +0+0−2+0−1+1 +0−1+1 +0−1+0 +0−1−1 1.100001101: +1.+1+1+0 +0−1−2 +0−2+1 +0−2+0+0−2−1 +0−2−1 +0−2−2 −1+1+1 −1+1+0 1.100001110: +1.+1+1+0 −1+1−1 −1+1−2−1+0+1 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1−1+1 1.100001111: +1.+1+1+0 −1−1+0−1−1−1 −1−1−1 −1−1−2 −1−2+1 −1−2+0 −1−2−1 −1−2−2 1.100010000: +1.+1+1+0−2+1+1 −2+1+1 −2+1+0 −2+1−1 −2+1−2 −2+0+1 −2+0+0 −2+0−1 1.100010001:+1.+1+1+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1−1 −2−1−2 −2−1−2 −2−2+11.100010010: +1.+1+1−1 +2−2+0 +2−2−1 +2−2−2 +1+1+1 +1+1+0 +1+1+0 +1+1−1+1+1−2 1.100010011: +1.+1+1−1 +1+0+1 +1+0+0 +1+0−1 +1+0−1 +1+0−2 +1−1+1+1−1+0 +1−1−1 1.100010100: +1.+1+1−1 +1−1−2 +1−2+1 +1−2+1 +1−2+0 +1−2−1+1−2−2 +0+1+1 +0+1+0 1.100010101: +1.+1+1−1 +0+1+0 +0+1−1 +0+1−2 +0+0+1+0+0+0 +0+0−1 +0+0−1 +0+0−2 1.100010110: +1.+1+1−1 +0−1+1 +0−1+0 +0−1−1+0−1−2 +0−2+1 +0−2+1 +0−2+0 +0−2−1 1.100010111: +1.+1+1−1 +0−2−2 −1+1+1−1+1+0 −1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+0 1.100011000: +1.+1+1−1 −1+0−1−1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1−1 −1−1−2 −1−1−2 1.100011001: +1.+1+1−1−1−2+1 −1−2+0 −1−2−1 −1−2−2 −2+1+1 −2+1+1 −2+1+0 −2+1−1 1.100011010:+1.+1+1−1 −2+1−2 −2+0+1 −2+0+0 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+01.100011011: +1.+1+1−1 −2−1−1 −2−1−1 −2−1−2 −2−2+1 −2−2+0 −2−2−1 −2−2−2−2−2−2 1.100011100: +1.+1+1−2 +1+1+1 +1+1+0 +1+1−1 +1+1−2 +1+1−2 +1+0+1+1+0+0 +1+0−1 1.100011101: +1.+1+1−2 +1+0−2 +1−1+1 +1−1+1 +1−1+0 +1−1−1+1−1−2 +1−2+1 +1−2+0 1.100011110: +1.+1+1−2 +1−2+0 +1−2−1 +1−2−2 +0+1+1+0+1+0 +0+1−1 +0+1−1 +0+1−2 1.100011111: +1.+1+1−2 +0+0+1 +0+0+0 +0+0−1+0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 Part 101.100100000000(1)-1.100111111111(1) 1.100100000: +1.+1+1−2 +0−1−2 +0−1−2+0−2+1 +0−2+0 +0−2−1 +0−2−2 +0−2−2 −1+1+1 1.100100001: +1.+1+1−2 −1+1+0−1+1−1 −1+1−2 −1+0+1 −1+0+1 −1+0+0 −1+0−1 −1+0−2 1.100100010: +1.+1+1−2−1−1+1 −1−1+1 −1−1+0 −1−1−1 −1−1−2 −1−2+1 −1−2+1 −1−2+0 1.100100011:+1.+1+1−2 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1+0 −2+1−1 −2+1−2 −2+0+11.100100100: +1.+1+1−2 −2+0+0 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1+0−2−1−1 1.100100101: +1.+1+0+1 +2−1−2 +2−2+1 +2−2+0 +2−2−1 +2−2−1 +2−2−2+1+1+1 +1+1+0 1.100100110: +1.+1+0+1 +1+1−1 +1+1−1 +1+1−2 +1+0+1 +1+0+0+1+0−1 +1+0−1 +1+0−2 1.100100111: +1.+1+0+1 +1−1+1 +1−1+0 +1−1−1 +1−1−1+1−1−2 +1−2+1 +1−2+0 +1−2−1 1.100101000: +1.+1+0+1 +1−2−1 +1−2−2 +0+1+1+0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1 1.100101001: +1.+1+0+1 +0+0+0 +0+0−1+0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 +0−1−1 1.100101010: +1.+1+0+1 +0−1−2+0−2+1 +0−2+0 +0−2−1 +0−2−1 +0−2−2 −1+1+1 −1+1+0 1.100101011: +1.+1+0+1−1+1−1 −1+1−1 −1+1−2 −1+0+1 −1+0+0 −1+0−1 −1+0−1 −1+0−2 1.100101100:+1.+1+0+1 −1−1+1 −1−1+0 −1−1−1 −1−1−1 −1−1−2 −1−2+1 −1−2+0 −1−2−11.100101101: +1.+1+0+1 −1−2−1 −1−2−2 −2+1+1 −2+1+0 −2+1−1 −2+1−1 −2+1−2−2+0+1 1.100101110: +1.+1+0+1 −2+0+0 −2+0+0 −2+0−1 −2+0−2 −2−1+1 −2−1+0−2−1+0 −2−1−1 1.100101111: +1.+1+0+0 +2−1−2 +2−2+1 +2−2+0 +2−2+0 +2−2−1+2−2−2 +1+1+1 +1+1+0 1.100110000: +1.+1+0+0 +1+1+0 +1+1−1 +1+1−2 +1+0+1+1+0+1 +1+0+0 +1+0−1 +1+0−2 1.100110001: +1.+1+0+0 +1−1+1 +1−1+1 +1−1+0+1−1−1 +1−1−2 +1−2+1 +1−2+1 +1−2+0 1.100110010: +1.+1+0+0 +1−2−1 +1−2−2+1−2−2 +0+1+1 +0+1+0 +0+1−1 +0+1−2 +0+1−2 1.100110011: +1.+1+0+0 +0+0+1+0+0+0 +0+0−1 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1−1 1.100110100: +1.+1+0+0+0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2−1 +0−2−1 +0−2−2 −1+1+1 1.100110101:+1.+1+0+0 −1+1+0 −1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+1 −1+0+0 −1+0−11.100110110: +1.+1+0+0 −1+0−2 −1−1+1 −1−1+1 −1−1+0 −1−1−1 −1−1−2 −1−1−2−1−2+1 1.100110111: +1.+1+0+0 −1−2+0 −1−2−1 −1−2−2 −1−2−2 −2+1+1 −2+1+0−2+1−1 −2+1−1 1.100111000: +1.+1+0+0 −2+1−2 −2+0+1 −2+0+0 −2+0−1 −2+0−1−2+0−2 −2−1+1 −2−1+0 1.100111001: +1.+1+0+0 −2−1+0 −2−1−1 −2−1−2 −2−2+1−2−2+1 −2−2+0 −2−2−1 −2−2−2 1.100111010: +1.+1+0−1 +1+1+1 +1+1+1 +1+1+0+1+1−1 +1+1−2 +1+1−2 +1+0+1 +1+0+0 1.100111011: +1.+1+0−1 +1+0−1 +1+0−1+1+0−2 +1−1+1 +1−1+0 +1−1−1 +1−1−1 +1−1−2 1.100111100: +1.+1+0−1 +1−2+1+1−2+0 +1−2+0 +1−2−1 +1−2−2 +0+1+1 +0+1+1 +0+1+0 1.100111101: +1.+1+0−1+0+1−1 +0+1−2 +0+1−2 +0+0+1 +0+0+0 +0+0−1 +0+0−1 +0+0−2 1.100111110:+1.+1+0−1 +0−1+1 +0−1+0 +0−1−1 +0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2+01.100111111: +1.+1+0−1 +0−2−1 +0−2−2 −1+1+1 −1+1+1 −1+1+0 −1+1−1 −1+1−2−1+1−2 Part 11 1.101000000000(1)-1.101011111111(1) 1.101000000:+1.+1+0−1 −1+0+1 −1+0+0 −1+0−1 −1+0−1 −1+0−2 −1−1+1 −1−1+0 −1−1+01.101000001: +1.+1+0−1 −1−1−1 −1−1−2 −1−2+1 −1−2+1 −1−2+0 −1−2−1 −1−2−2−1−2−2 1.101000010: +1.+1+0−1 −2+1+1 −2+1+0 −2+1−1 −2+1−1 −2+1−2 −2+0+1−2+0+0 −2+0−1 1.101000011: +1.+1+0−1 −2+0−1 −2+0−2 −2−1+1 −2−1+0 −2−1+0−2−1−1 −2−1−2 −2−2+1 1.101000100: +1.+1+0−2 +2−2+1 +2−2+0 +2−2−1 +2−2−2+2−2−2 +1+1+1 +1+1+0 +1+1−1 1.101000101: +1.+1+0−2 +1+1−1 +1+1−2 +1+0+1+1+0+0 +1+0+0 +1+0−1 +1+0−2 +1+0−2 1.101000110: +1.+1+0−2 +1−1+1 +1−1+0+1−1−1 +1−1−1 +1−1−2 +1−2+1 +1−2+0 +1−2+0 1.101000111: +1.+1+0−2 +1−2−1+1−2−2 +0+1+1 +0+1+1 +0+1+0 +0+1−1 +0+1−2 +0+1−2 1.101001000: +1.+1+0−2+0+0+1 +0+0+0 +0+0−1 +0+0−1 +0+0−2 +0−1+1 +0−1+0 +0−1+0 1.101001001:+1.+1+0−2 +0−1−1 +0−1−2 +0−2+1 +0−2+1 +0−2+0 +0−2−1 +0−2−2 +0−2−21.101001010: +1.+1+0−2 −1+1+1 −1+1+0 −1+1+0 −1+1−1 −1+1−2 −1+0+1 −1+0+1−1+0+0 1.101001011: +1.+1+0−2 −1+0−1 −1+0−2 −1+0−2 −1−1+1 −1−1+0 −1−1−1−1−1−1 −1−1−2 1.101001100: +1.+1+0−2 −1−2+1 −1−2+0 −1−2+0 −1−2−1 −1−2−2−1−2−2 −2+1+1 −2+1+0 1.101001101: +1.+1+0−2 −2+1−1 −2+1−1 −2+1−2 −2+0+1−2+0+0 −2+0+0 −2+0−1 −2+0−2 1.101001110: +1.+1+0−2 −2−1+1 −2−1+1 −2−1+0−2−1−1 −2−1−1 −2−1−2 −2−2+1 −2−2+0 1.101001111: +1.+1−1+1 +2−2+0 +2−2−1+2−2−2 +1+1+1 +1+1+1 +1+1+0 +1+1−1 +1+1−2 1.101010000: +1.+1−1+1 +1+1−2+1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−2 +1−1+1 +1−1+1 1.101010001: +1.+1−1+1+1−1+0 +1−1−1 +1−1−2 +1−1−2 +1−2+1 +1−2+0 +1−2+0 +1−2−1 1.101010010:+1.+1−1+1 +1−2−2 +0+1+1 +0+1+1 +0+1+0 +0+1−1 +0+1−2 +0+1−2 +0+0+11.101010011: +1.+1−1+1 +0+0+0 +0+0+0 +0+0−1 +0+0−2 +0−1+1 +0−1+1 +0−1+0+0−1−1 1.101010100: +1.+1−1+1 +0−1−1 +0−1−2 +0−2+1 +0−2+0 +0−2+0 +0−2−1+0−2−2 −1+1+1 1.101010101: +1.+1−1+1 −1+1+1 −1+1+0 −1+1−1 −1+1−1 −1+1−2−1+0+1 −1+0+0 −1+0+0 1.101010110: +1.+1−1+1 −1+0−1 −1+0−2 −1+0−2 −1−1+1−1−1+0 −1−1−1 −1−1−1 −1−1−2 1.101010111: +1.+1−1+1 −1−2+1 −1−2+1 −1−2+0−1−2−1 −1−2−2 −1−2−2 −2+1+1 −2+1+0 1.101011000: +1.+1−1+1 −2+1+0 −2+1−1−2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0−1 −2+0−1 1.101011001: +1.+1−1+1 −2+0−2−2−1+1 −2−1+0 −2−1+0 −2−1−1 −2−1−2 −2−1−2 −2−2+1 1.101011010: +1.+1−1+0+2−2+0 +2−2−1 +2−2−1 +2−2−2 +1+1+1 +1+1+1 +1+1+0 +1+1−1 1.101011011:+1.+1−1+0 +1+1−2 +1+1−2 +1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−2 +1−1+11.101011100: +1.+1−1+0 +1−1+1 +1−1+0 +1−1−1 +1−1−1 +1−1−2 +1−2+1 +1−2+0+1−2+0 1.101011101: +1.+1−1+0 +1−2−1 +1−2−2 +1−2−2 +0+1+1 +0+1+0 +0+1+0+0+1−1 +0+1−2 1.101011110: +1.+1−1+0 +0+0+1 +0+0+1 +0+0+0 +0+0−1 +0+0−1+0+0−2 +0−1+1 +0−1+0 1.101011111: +1.+1−1+0 +0−1+0 +0−1−1 +0−1−2 +0−1−2+0−2+1 +0−2+0 +0−2+0 +0−2−1 Part 12 1.101100000000(1)-1.101111111111(1)1.101100000: +1.+1−1+0 +0−2−2 −1+1+1 −1+1+1 −1+1+0 −1+1−1 −1+1−1 −1+1−2−1+0+1 1.101100001: +1.+1−1+0 −1+0+1 −1+0+0 −1+0−1 −1+0−2 −1+0−2 −1−1+1−1−1+0 −1−1+0 1.101100010: +1.+1−1+0 −1−1−1 −1−1−2 −1−1−2 −1−2+1 −1−2+0−1−2−1 −1−2−1 −1−2−2 1.101100011: +1.+1−1+0 −2+1+1 −2+1+1 −2+1+0 −2+1−1−2+1−1 −2+1−2 −2+0+1 −2+0+0 1.101100100: +1.+1−1+0 −2+0+0 −2+0−1 −2+0−2−2+0−2 −2−1+1 −2−1+0 −2−1+0 −2−1−1 1.101100101: +1.+1−1−1 +2−1−2 +2−1−2+2−2+1 +2−2+0 +2−2−1 +2−2−1 +2−2−2 +1+1+1 1.101100110: +1.+1−1−1 +1+1+1+1+1+0 +1+1−1 +1+1−1 +1+1−2 +1+0+1 +1+0+1 +1+0+0 1.101100111: +1.+1−1−1+1+0−1 +1+0−2 +1+0−2 +1−1+1 +1−1+0 +1−1+0 +1−1−1 +1−1−2 1.101101000:+1.+1−1−1 +1−1−2 +1−2+1 +1−2+0 +1−2+0 +1−2−1 +1−2−2 +1−2−2 +0+1+11.101101001: +1.+1−1−1 +0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1 +0+0+1 +0+0+0+0+0−1 1.101101010: +1.+1−1−1 +0+0−1 +0+0−2 +0−1+1 +0−1+1 +0−1+0 +0−1−1+0−1−1 +0−1−2 1.101101011: +1.+1−1−1 +0−2+1 +0−2+0 +0−2+0 +0−2−1 +0−2−2+0−2−2 −1+1+1 −1+1+0 1.101101100: +1.+1−1−1 −1+1+0 −1+1−1 −1+1−2 −1+1−2−1+0+1 −1+0+0 −1+0+0 −1+0−1 1.101101101: +1.+1−1−1 −1+0−2 −1+0−2 −1−1+1−1−1+0 −1−1−1 −1−1−1 −1−1−2 −1−2+1 1.101101110: +1.+1−1−1 −1−2+1 −1−2+0−1−2−1 −1−2−1 −1−2−2 −2+1+1 −2+1+1 −2+1+0 1.101101111: +1.+1−1−1 −2+1−1−2+1−1 −2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0−1 −2+0−1 1.101110000: +1.+1−1−1−2+0−2 −2−1+1 −2−1+1 −2−1+0 −2−1−1 −2−1−1 −2−1−2 −2−2+1 1.101110001:+1.+1−1−2 +2−2+1 +2−2+0 +2−2−1 +2−2−2 +2−2−2 +1+1+1 +1+1+0 +1+1+01.101110010: +1.+1−1−2 +1+1−1 +1+1−2 +1+1−2 +1+0+1 +1+0+0 +1+0+0 +1+0−1+1+0−2 1.101110011: +1.+1−1−2 +1+0−2 +1−1+1 +1−1+0 +1−1+0 +1−1−1 +1−1−2+1−1−2 +1−2+1 1.101110100: +1.+1−1−2 +1−2+0 +1−2+0 +1−2−1 +1−2−2 +1−2−2+0+1+1 +0+1+0 +0+1+0 1.101110101: +1.+1−1−2 +0+1−1 +0+1−2 +0+1−2 +0+0+1+0+0+0 +0+0+0 +0+0−1 +0+0−2 1.101110110: +1.+1−1−2 +0+0−2 +0−1+1 +0−1+0+0−1+0 +0−1−1 +0−1−2 +0−1−2 +0−2+1 1.101110111: +1.+1−1−2 +0−2+0 +0−2+0+0−2−1 +0−2−2 +0−2−2 −1+1+1 −1+1+0 −1+1+0 1.101111000: +1.+1−1−2 −1+1−1−1+1−2 −1+1−2 −1+0+1 −1+0+0 −1+0+0 −1+0−1 −1+0−2 1.101111001: +1.+1−1−2−1+0−2 −1−1+1 −1−1+0 −1−1+0 −1−1−1 −1−1−2 −1−1−2 −1−2+1 1.101111010:+1.+1−1−2 −1−2+0 −1−2+0 −1−2−1 −1−2−2 −1−2−2 −2+1+1 −2+1+0 −2+1+01.101111011: +1.+1−1−2 −2+1−1 −2+1−2 −2+1−2 −2+0+1 −2+0+0 −2+0+0 −2+0−1−2+0−2 1.101111100: +1.+1−1−2 −2+0−2 −2−1+1 −2−1+0 −2−1+0 −2−1−1 −2−1−1−2−1−2 −2−2+1 1.101111101: +1.+1−2+1 +2−2+1 +2−2+0 +2−2−1 +2−2−1 +2−2−2+1+1+1 +1+1+1 +1+1+0 1.101111110: +1.+1−2+1 +1+1−1 +1+1−1 +1+1−2 +1+0+1+1+0+1 +1+0+0 +1+0−1 +1+0−1 1.101111111: +1.+1−2+1 +1+0−2 +1−1+1 +1−1+1+1−1+0 +1−1−1 +1−1−1 +1−1−2 +1−2+1 Part 131.110000000000(1)-1.110011111111(1) 1.110000000: +1.+1−2+1 +1−2+1 +1−2+0+1−2+0 +1−2−1 +1−2−2 +1−2−2 +0+1+1 +0+1+0 1.110000001: +1.+1−2+1 +0+1+0+0+1−1 +0+1−2 +0+1−2 +0+0+1 +0+0+0 +0+0+0 +0+0−1 1.110000010: +1.+1−2+1+0+0−2 +0+0−2 +0−1+1 +0−1+0 +0−1+0 +0−1−1 +0−1−2 +0−1−2 1.110000011:+1.+1−2+1 +0−2+1 +0−2+1 +0−2+0 +0−2−1 +0−2−1 +0−2−2 −1+1+1 −1+1+11.110000100: +1.+1−2+1 −1+1+0 −1+1−1 −1+1−1 −1+1−2 −1+0+1 −1+0+1 −1+0+0−1+0−1 1.110000101: +1.+1−2+1 −1+0−1 −1+0−2 −1+0−2 −1−1+1 −1−1+0 −1−1+0−1−1−1 −1−1−2 1.110000110: +1.−1−2+1 −1−1−2 −1−2+1 −1−2+0 −1−2+0 −1−2−1−1−2−2 −1−2−2 −2+1+1 1.110000111: +1.+1−2+1 −2+1+1 −2+1+0 −2+1−1 −2+1−1−2+1−2 −2+0+1 −2+0+1 −2+0+0 1.110001000: +1.+1−2+1 −2+0−1 −2+0−1 −2+0−2−2−1+1 −2−1+1 −2−1+0 −2−1+0 −2−1−1 1.110001001: +1.+1−2+1 −2−1−2 −2−1−2−2−2+1 −2−2+0 −2−2+0 −2−2−1 −2−2−2 −2−2−2 1.110001010: +1.+1−2+0 +1+1+1+1+1+1 +1+1+0 +1+1−1 +1+1−1 +1+1−2 +1+0+1 +1+0+1 1.110001011: +1.+1−2+0+1+0+0 +1+0−1 +1+0−1 +1+0−2 +1+0−2 +1−1+1 +1−1+0 +1−1+0 1.110001100:+1.+1−2+0 +1−1−1 +1−1−2 +1−1−2 +1−2+1 +1−2+0 +1−2+0 +1−2−1 +1−2−11.110001101: +1.+1−2+0 +1−2−2 +0+1+1 +0+1+1 +0+1+0 +0+1−1 +0+1−1 +0+1−2+0+0+1 1.110001110: +1.+1−2+0 +0+0+1 +0+0+0 +0+0+0 +0+0−1 +0+0−2 +0+0−2+0−1+1 +0−1+0 1.110001111: +1.+1−2+0 +0−1+0 +0−1−1 +0−1−2 +0−1−2 +0−2−2+0−2+1 +0−2+0 +0−2−1 1.110010000: +1.+1−2+0 +0−2−1 +0−2−2 −1+1+1 −1+1+1−1+1+0 −1+1+0 −1+1−1 −1+1−2 1.110010001: +1.+1−2+0 −1+1−2 −1+0+1 −1+0+0−1+0+0 −1+0−1 −1+0−1 −1+0−2 −1−1+1 1.110010010: +1.+1−2+0 −1−1+1 −1−1+0−1−1−1 −1−1−1 −1−1−2 −1−1−2 −1−2+1 −1−2+0 1.110010011: +1.+1−2+0 −1−2+0−1−2−1 −1−2−2 −1−2−2 −2+1+1 −2+1+0 −2+1+0 −2+1−1 1.110010100: +1.+1−2+0−2+1−1 −2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0+0 −2+0−1 −2+0−2 1.110010101:+1.+1−2+0 −2+0−2 −2−1+1 −2−1+0 −2−1+0 −2−1−1 −2−1−1 −2−1−2 −2−2+11.110010110: +1.+1−2−1 +2−2+1 +2−2+0 +2−2−1 +2−2−1 +2−2−2 +2−2−2 +1+1+1+1+1+0 1.110010111: +1.+1−2−1 +1+1+0 +1+1−1 +1+1−2 +1+1−2 +1+0+1 +1+0+1+1+0+0 +1+0−1 1.110011000: +1.+1−2−1 +1+0−1 +1+0−2 +1−1+1 +1−1+1 +1−1+0+1−1+0 +1−1−1 +1−1−2 1.110011001: +1.+1−2−1 +1−1−2 +1−2+1 +1−2+1 +1−2+0+1−2−1 +1−2−1 +1−2−2 +0+1+1 1.110011010: +1.+1−2−1 +0+1+1 +0+1+0 +0+1+0+0+1−1 +0+1−2 +0+1−2 +0+0+1 +0+0+1 1.110011011: +1.+1−2−1 +0+0+0 +0+0−1+0+0−1 +0+0−2 +0−1+1 +0−1+1 +0−1+0 +0−1+0 1.110011100: +1.+1−2−1 +0−1−1+0−1−2 +0−1−2 +0−2+1 +0−2+1 +0−2+0 +0−2−1 +0−2−1 1.110011101: +1.+1−2−1+0−2−2 −1+1+1 −1+1+1 −1+1+0 −1+1+0 −1+1−1 −1+1−2 −1+1−2 1.110011110:+1.+1−2−1 −1+0+1 −1+0+1 −1+0+0 −1+0−1 −1+0−1 −1+0−2 −1+0−2 −1−1+11.110011111: +1.+1−2−1 −1−1+0 −1−1+0 −1−1−1 −1−1−2 −1−1−2 −1−2+1 −1−2+1−1−2+0 Part 14 1.110100000000(1)-1.110111111111(1) 1.110100000:+1.+1−2−1 −1−2−1 −1−2−1 −1−2−2 −1−2−2 −2+1+1 −2+1+0 −2+1+0 −2+1−11.110100001: +1.+1−2−1 −2+1−1 −2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0+0 −2+0−1−2+0−2 1.110100010: +1.+1−2−1 −2+0−2 −2−1+1 −2−1+0 −2−1+0 −2−1−1 −2−1−1−2−1−2 −2−2+1 1.110100011: +1.+1−2−2 +2−2+1 +2−2+0 +2−2+0 +2−2−1 +2−2−2+2−2−2 +1+1+1 +1+1+1 1.110100100: +1.+1−2−2 +1+1+0 +1+1−1 +1+1−1 +1+1−2+1+1−2 +1+0+1 +1+0+0 +1+0+0 1.110100101: +1.+1−2−2 +1+0−1 +1+0−1 +1+0−2+1−1+1 +1−1+1 +1−1+0 +1−1+0 +1−1−1 1.110100110: +1.+1−2−2 +1−1−2 +1−1−2+1−2+1 +1−2+1 +1−2+0 +1−2−1 +1−2−1 +1−2−2 1.110100111: +1.+1−2−2 +1−2−2+0+1+1 +0+1+0 +0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1 1.110101000: +1.+1−2−2+0+0+1 +0+0+0 +0+0+0 +0+0−1 +0+0−2 +0+0−2 +0−1+1 +0−1+1 1.110101001:+1.+1−2−2 +0−1+0 +0−1−1 +0−1−1 +0−1−2 +0−1−2 +0−2+1 +0−2+0 +0−2+01.110101010: +1.+1−2−2 +0−2−1 +0−2−1 +0−2−2 −1+1+1 −1+1+1 −1+1+0 −1+1+0−1+1−1 1.110101011: +1.+1−2−2 −1+1−2 −1+1−2 −1+0+1 −1+0+1 −1+0+0 −1+0+0−1+0−1 −1+0−2 1.110101100: +1.+1−2−2 −1+0−2 −1−1+1 −1−1+1 −1−1+0 −1−1−1−1−1−1 −1−1−2 −1−1−2 1.110101101: +1.+1−2−2 −1−2+1 −1−2+0 −1−2+0 −1−2−1−1−2−1 −1−2−2 −2+1+1 −2+1+1 1.110101110: +1.+1−2−2 −2+1+0 −2+1+0 −2+1−1−2+1−2 −2+1−2 −2+0+1 −2+0+1 −2+0+0 1.110101111: +1.+1−2−2 −2+0+0 −2+0−1−2+0−2 −2+0−2 −2−1+1 −2−1+1 −2−1+0 −2−1−1 1.110110000: +1.+1−2−2 −2−1−1−2−1−2 −2−1−2 −2−2+1 −2−2+0 −2−2+0 −2−2−1 −2−2−1 1.110110001: +1.+0+1+1+2−2−2 +2−2−2 +1+1+1 +1+1+0 +1+1+0 +1+1−1 +1+1−1 +1+1−2 1.110110010:+1.+0+1+1 +1+0+1 +1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−1 +1+0−2 +1−1+11.110110011: +1.+0+1+1 +1−1+1 +1−1+0 +1−1+0 +1−1−1 +1−1−2 +1−1−2 +1−2+1+1−2+1 1.110110100: +1.+0+1+1 +1−2+0 +1−2−1 +1−2−1 +1−2−2 +1−2−2 +0+1+1+0+1+1 +0+1+0 1.110110101: +1.+0+1+1 +0+1−1 +0+1−1 +0+1−2 +0+1−2 +0+0+1+0+0+1 +0+0+0 +0+0−1 1.110110110: +1.+0+1+1 +0+0−1 +0+0−2 +0+0−2 +0−1+1+0−1+0 +0−1+0 +0−1−1 +0−1−1 1.110110111: +1.+0+1+1 +0−1−2 +0−1−2 +0−2+1+0−2+0 +0−2+0 +0−2−1 +0−2−1 +0−2−2 1.110111000: +1.+0+1+1 −1+1+1 −1+1+1−1+1+0 −1+1+0 −1+1−1 −1+1−1 −1+1−2 −1+0+1 1.110111001: +1.+0+1+1 −1+0+1−1+0+0 −1+0+0 −1+0−1 −1+0−1 −1+0−2 −1−1+1 −1−1+1 1.110111010: +1.+0+1+1−1−1+0 −1−1+0 −1−1−1 −1−1−1 −1−1−2 −1−2+1 −1−2+1 −1−2+0 1.110111011:+1.+0+1+1 −1−2+0 −1−2−1 −1−2−1 −1−2−2 −2+1+1 −2+1+1 −2+1+0 −2+1+01.110111100: +1.+0+1+1 −2+1−1 −2+1−2 −2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0+0−2+0−1 1.110111101. +1.+0+1+1 −2+0−2 −2+0−2 −2−1+1 −2−1+1 −2−1+0 −2−1+0−2−1−1 −2−1−2 1.110111110: +1.+0+1+1 −2−1−2 −2−2+1 −2−2+1 −2−2+0 −2−2+0−2−2−1 −2−2−2 −2−2−2 1.110111111: +1.+0+1+0 +1+1+1 +1+1+1 +1+1+0 +1+1+0+1+1−1 +1+1−2 +1+1−2 +1+0+1 Part 15 1.111000000000(1)-1.111011111111(1)1.111000000: +1.+0+1+0 +1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−1 +1+0−2 +1−1+1+1−1+1 1.111000001: +1.+0+1+0 +1−1+0 +1−1+0 +1−1−1 +1−1−1 +1−1−2 +1−2+1+1−2+1 +1−2+0 1.111000010: +1.+0+1+0 +1−2+0 +1−2−1 +1−2−1 +1−2−2 +0+1+1+0+1+1 +0+1+0 +0+1+0 1.111000011: +1.+0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1+0+0+1 +0+0+0 +0+0+0 +0+0−1 1.111000100: +1.+0+1+0 +0+0−1 +0−0−2 +0+0−2+0−1+1 +0−1+0 +0−1+0 +0−1−1 +0−1−1 1.111000101: +1.+0+1+0 +0−1−2 +0−1−2+0−2+1 +0−2+0 +0−2+0 +0−2−1 +0−2−1 +0−2−2 1.111000110: +1.+0+1+0 +0−2−2−1+1+1 −1+1+1 −1+1+0 −1+1−1 −1+1−1 −1+1−2 −1+1−2 1.111000111: +1.+0+1+0−1+0+1 −1+0+1 −1+0+0 −1+0−1 −1+0−1 −1+0−2 −1+0−2 −1−1+1 1.111001000:+1.+0+1+0 −1−1+1 −1−1+0 −1−1+0 −1−1−1 −1−1−2 −1−1−2 −1−2+1 −1−2+11.111001001: +1.+0+1+0 −1−2+0 −1−2+0 −1−2−1 −1−2−1 −1−2−2 −2+1+1 −2+1+1−2+1+0 1.111001010: +1.+0+1+0 −2+1+0 −2+1−1 −2+1−1 −2+1−2 −2+1−2 −2+0+1−2+0+0 −2+0+0 1.111001011: +1.+0+1+0 −2+0−1 −2+0−1 −2+0−2 −2+0−2 −2−1+1−2−1+1 −2−1+0 −2−1−1 1.111001100: +1.+0+1+0 −2−1−1 −2−1−2 −2−1−2 −2−2+1−2−2+1 −2−2+0 −2−2+0 −2−2−1 1.111001101: +1.+0+1−1 +2−2−2 +2−2−2 +1+1+1+1+1+1 +1+1+0 +1+1+0 +1+1−1 +1+1−1 1.111001110: +1.+0+1−1 +1+1−2 +1+0+1+1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−1 +1+0−2 1.111001111: +1.+0+1−1 +1+0−2+1−1+1 +1−1+0 +1−1+0 +1−1−1 +1−1−1 +1−1−2 +1−1−2 1.111010000: +1.+0+1−1+1−2+1 +1−2+1 +1−2+0 +1−2+0 +1−2−1 +1−2−2 +1−2−2 +0+1+1 1.111010001:+1.+0+1−1 +0+1+1 +0+1+0 +0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+0+1 +0+0+11.111010010: +1.+0+1−1 +0+0+0 +0+0+0 +0+0−1 +0+0−1 +0+0−2 +0+0−2 +0−1+1+0−1+1 1.111010011: +1.+0+1−1 +0−1+0 +0−1−1 +0−1−1 +0−1−2 +0−1−2 +0−2+1+0−2+1 +0−2+0 1.111010100: +1.+0+1−1 +0−2+0 +0−2−1 +0−2−1 +0−2−2 −1+1+1−1+1+1 −1+1+0 −1+1+0 1.111010101: +1.+0+1−1 −1+1−1 −1+1−1 −1+1−2 −1+1−2−1+0+1 −1+0+1 −1+0+0 −1+0−1 1.111010110: +1.+0+1−1 −1+0−1 −1+0−2 −1+0−2−1−1+1 −1−1+1 −1−1−0 −1−1+0 −1−1−1 1.111010111: +1.+0+1−1 −1−1−1 −1−1−2−1−2+1 −1−2+1 −1−2+0 −1−2+0 −1−2−1 −1−2−1 1.111011000: +1.+0+1−1 −1−2−2−1−2−2 −2+1+1 −2+1+1 −2+1+0 −2+1+0 −2+1−1 −2+1−2 1.111011001: +1.+0+1−1−2+1−2 −2+0+1 −2+0+1 −2+0+0 −2+0+0 −2+0−1 −2+0−1 −2+0−2 1.111011010:+1.+0+1−1 −2+0−2 −2−1+1 −2−1+1 −2−1+0 −2−1−1 −2−1−1 −2−1−2 −2−1−21.111011011: +1.+0+1−1 −2−2+1 −2−2+1 −2−2+0 −2−2+0 −2−2−1 −2−2−1 −2−2−2−2−2−2 1.111011100: +1.+0+1−2 +1+1+1 +1+1+0 +1+1+0 +1+1−1 +1+1−1 +1+1−2+1+1−2 +1+0+1 1.111011101: +1.+0+1−2 +1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−1+1+0−2 +1−1+1 +1−1+1 1.111011110: +1.+0+1−2 +1−1+0 +1−1+0 +1−1−1 +1−1−1+1−1−2 +1−1−2 +1−2+1 +1−2+1 1.111011111: +1.+0+1−2 +1−2+0 +1−2+0 +1−2−1+1−2−1 +1−2−2 +0+1+1 +0+1+1 +0+1+0 Part 161.111100000000(1)-1.111111111111(1) 1.111100000: +1.+0+1−2 +0+1+0 +0+1−1+0+1−1 +0+1−2 +0+1−2 +0+0+1 +0+0+1 +0+0+0 1.111100001: +1.+0+1−2 +0+0+0+0+0−1 +0+0−1 +0+0−2 +0−1+1 +0−1+1 +0−1+0 +0−1+0 1.111100010: +1.+0+1−2+0−1−1 +0−1−1 +0−1−2 +0−1−2 +0−2+1 +0−2+1 +0−2+0 +0−2+0 1.111100011:+1.+0+1−2 +0−2−1 +0−2−1 +0−2−2 +0−2−2 −1+1+1 −1+1+0 −1+1+0 −1+1−11.111100100: +1.+0+1−2 −1+1−1 −1+1−2 −1+1−2 −1+0+1 −1+0+1 −1+0+0 −1+0+0−1+0−1 1.111100101: +1.+0+1−2 −1+0−1 −1+0−2 −1+0−2 −1−1+1 −1−1+1 −1−1+0−1−1−1 −1−1−1 1.111100110: +1.+0+1−2 −1−1−2 −1−1−2 −1−2+1 −1−2+1 −1−2+0−1−2+0 −1−2−1 −1−2−1 1.111100111: +1.+0+1−2 −1−2−2 −1−2−2 −2+1+1 −2+1+1−2+1+0 −2+1+0 −2+1−1 −2+1−1 1.111101000: +1.+0+1−2 −2+1−2 −2+1−2 −2+0+1−2+0+0 −2+0+0 −2+0−1 −2+0−1 −2+0−2 1.111101001: +1.+0+1−2 −2+0−2 −2−1+1−2−1+1 −2−1+0 −2−1+0 −2−1−1 −2−1−1 −2−1−2 1.111101010: +1.+0+1−2 −2−1−2−2−2+1 −2−2+1 −2−2+0 −2−2+0 −2−2−1 −2−2−1 −2−2−2 1.111101011: +1.+0+0+1+1+1+1 +1+1+1 +1+1+0 +1+1+0 +1+1−1 +1+1−1 +1+1−2 +1+1−2 1.111101100:+1.+0+0+1 +1+0+1 +1+0+1 +1+0+0 +1+0+0 +1+0−1 +1+0−1 +1+0−2 +1+0−21.111101101: +1.+0+0+1 +1−1+1 +1−1+1 +1−1+0 +1−1+0 +1−1−1 +1−1−1 +1−1−2+1−1−2 1.111101110: +1.+0+0+1 +1−2+1 +1−2+1 +1−2+0 +1−2−1 +1−2−1 +1−2−2+1−2−2 +0+1+1 1.111101111: +1.+0+0+1 +0+1+1 +0+1+0 +0+1+0 +0+1−1 +0+1−1+0+1−2 +0+1−2 +0+0+1 1.111101000: +1.+0+0+1 +0+0+1 +0+0+0 +0+0+0 +0+0−1+0+0−1 +0+0−2 +0+0−2 +0−1+1 1.111110001: +1.+0+0+1 +0−1+1 +0−1+0 +0−1+0+0−1−1 +0−1−1 +0−1−2 +0−1−2 +0−2+1 1.111110010: +1.+0+0+1 +0−2+1 +0−2+0+0−2−1 +0−2−1 +0−2−2 +0−2−2 −1+1+1 −1+1+1 1.111110011: +1.+0+0+1 −1+1+0−1+1+0 −1+1−1 −1+1−1 −1+1−2 −1+1−2 −1+0+1 −1+0+1 1.111110100: +1.+0+0+1−1+0+0 −1+0+0 −1+0−1 −1+0−1 −1+0−2 −1+0−2 −1−1+1 −1−1+1 1.111110101:+1.+0+0+1 −1−1+0 −1−1+0 −1−1−1 −1−1−1 −1−1−2 −1−1−2 −1−2+1 −1−2+11.111110110: +1.+0+0+1 −1−2+0 −1−2+0 −1−2−1 −1−2−1 −1−2−2 −1−2−2 −2+1+1−2+1+1 1.111110111: +1.+0+0+1 −2+1+0 −2+1+0 −2+1−1 −2+1−1 −2+1−2 −2+1−2−2+0+1 −2+0+1 1.111111000: +1.+0+0+1 −2+0+0 −2+0−1 −2+0−1 −2+0−2 −2+0−2−2−1+1 −2−1+1 −2−1+0 1.111111001: +1.+0+0+1 −2−1+0 −2−1−1 −2−1−1 −2−1−2−2−1−2 −2−2+1 −2−2+1 −2−2+0 1.111111010: +1.+0+0+0 +2−2+0 +2−2−1 +2−2−1+2−2−2 +2−2−2 +1+1+1 +1+1+1 +1+1+0 1.111111011. +1.+0+0+0 +1+1+0 +1+1−1+1+1−1 +1+1−2 +1+1−2 +1+0+1 +1+0+1 +1+0+0 1.111111100: +1.+0+0+0 +1+0+0+1+0−1 +1+0−1 +1+0−2 +1+0−2 +1−1+1 +1−1+1 +1−1+0 1.111111101: +1.+0+0+0+1−1+0 +1−1−1 +1−1−1 +1−1−2 +1−1−2 +1−2+1 +1−2+1 +1−2+0 1.111111110:+1.+0+0+0 +1−2+0 +1−2−1 +1−2−1 +1−2−2 +1−2−2 +0+1+1 +0+1+1 +0+1+01.111111111: +1.+0+0+0 +0+1+0 +0+1−1 +0+1−1 +0+1−2 +0+1−2 +0+0+1 +0+0+1+0+0+0

APPENDIX THREE 000 001 010 011 100 101 110 111 There are no Parts 1−14.Part 15 [Alternate Version] 1.111000000000(1)-1.111011111111(1)1.111000000: +1.+0+1+0+1 +0+1 +0+0 +0+0 +0−1 +0−1 +0−2 −1+1 −1+11.111000001: +1.+0+1+0+1 −1+0 −1+0 −1−1 −1−1 −1−2 −2+1 −2+1 −2+01.111000010: +1.+0+1+0+0 +2+0 +2−1 +2−1 +2−2 +1+1 +1+1 +1+0 +1+01.111000011: +1.+0+1+0+0 +1−1 +1−1 +1−2 +0+1 +0+1 +0+0 +0+0 −0−11.111000100: +1.+0+1+0+0 +0−1 +0−2 +0−2 −1+1 −1+0 −1+0 −1−1 −1−11.111000101: +1.+0+1+0+0 −1−2 −1−2 −2+1 −2+0 −2+0 −2−1 −2−1 −2−21.111000110: +1.+0+1+0−1 +2−2 +1+1 +1+1 +1+0 +1−1 +1−1 +1−2 +1−21.111000111: +1.+0+1+0−1 +0+1 +0+1 +0+0 +0−1 +0−1 +0−2 +0−2 −1+11.111001000: +1.+3+1+0−1 −1+1 −1+0 −1+0 −1−1 −1−2 −1−2 −2+1 −2+11.111001001: +1.+0+1+0−2 +2+0 +2+0 +2−1 +2−1 +2−2 +1+1 +1+1 +1+01.111001010: +1.+0+1+0−2 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+0 +0+01.111001011: +1.+0+1+0−2 +0−1 +0−1 +0−2 +0−2 −1+1 −1+1 −1+0 −1−11.111001100: +1.+0+1+0−2 −1−1 −1−2 −1−2 −2+1 −2+1 −2+0 −2+0 −2−11.111001101: +1.+0+1−1+1 +2−2 +2−2 +1+1 +1+1 +1+0 +1+0 +1−1 +1−11.111001110: +1.+0+1−1+1 +1−2 +0+1 +0+1 +0+0 +0+0 +0−1 +0−1 +0−21.111001111: +1.+0+1−1+1 +0−2 −1+1 −1+0 −1+0 −1−1 −1−1 −1−2 −1−21.111010000: +1.+0+1−1+0 +2+1 +2+1 +2+0 +2+0 +2−1 +2−2 +2−2 +1+11.111010001: +1.+0+1−1+0 +1+1 +1+0 +1+0 +1−1 +1−1 +1−2 +0+1 +0+11.111010010: +1.+0+1−1+0 +0+0 +0+0 +0−1 +0−1 +0−2 +0−2 −1+1 −1+11.111010011: +1.+0+1−1+0 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+1 −2+01.111010100: +1.+0+1−1−1 +2+0 +2−1 +2−1 +2−2 +1+1 +1+1 +1+0 +1+01.111010101: +1.+0+1−1−1 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 +0+0 +0−11.111010110: +1.+0+1−1−1 +0−1 +0−2 +0−2 −1+1 −1+1 −1+0 −1+0 −1−11.111010111: +1.+0+1−1−1 −1−1 −1−2 −2+1 −2+1 −2+0 −2+0 −2−1 −2−11.111011000: +1.+0+1−1−2 +2−2 +2−2 +1+1 +1+1 +1+0 +1+0 +1−1 +1−21.111011001: +1.+0+1−1−2 +1−2 +0+1 +0+1 +0+0 +0+0 +0−1 +0−1 +0−21.111011010: +1.+0+1−1−2 +0−2 −1+1 −1+1 −1+0 −1−1 −1−1 −1−2 −1−21.111011011: +1.+0+1−1−2 −2+1 −2+1 −2+0 −2+0 −2−1 −2−1 −2−2 −2−21.111011100: +1.+0+1−2+1 +1+1 +1+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+11.111011101: +1.+0+1−2+1 +0+1 +0+0 +0+0 +0−1 +0−1 +0−2 −1+1 −1+11.111011110: +1.+9+1−2+1 −1+0 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+11.111011111: +1.+0+1−2+0 +2+0 +2+0 +2−1 +2−1 +2−2 +1+1 +1+1 +1+0 Part 16[Alternate Version] 1.111000000000(1)-1.111011111111(1) 1.111100000:+1.+0+1−2+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 +0+0 1.111100001:+1.+0+1−2+0 +0+0 +0−1 +0−1 +0−2 −1+1 −1+1 −1+0 −1+0 1.111100010:+1.+0+1−2+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+1 −2+0 −2+0 1.111100011:+1.+0+1−2−1 +2−1 +2−1 +2−2 +2−2 +1+1 +1+0 +1+0 +1−1 1.111100100:+1.+0+1−2−1 +1−1 +1−2 +1−2 +0+1 +0+1 +0+0 +0+0 +0−1 1.111100101:+1.+0+1−2−1 +0−1 +0−2 +0−2 −1+1 −1+1 −1+0 −1−1 −1−1 1.111100110:+1.+0+1−2−1 −1−2 −1−2 −2+1 −2+1 −2+0 −2+0 −2−1 −2−1 1.111100111:+1.+0+1−2−2 +2−2 +2−2 +1+1 +1+1 +1+0 +1+0 +1−1 +1−1 1.111101000:+1.+0+1−2−2 +1−2 +1−2 +0+1 +0+0 +0+0 +0−1 +0−1 +0−2 1.111101001:+1.+0+1−2−2 +0−2 −1+1 −1+1 −1+0 −1+0 −1−1 −1−1 −1−2 1.111101010:+1.+0+1−2−2 −1−2 −2+1 −2+1 −2+0 −2+0 −2−1 −2−1 −2−2 1.111101011:+1.+0+0+1+1 +1+1 +1+1 +1+0 +1+0 +1−1 +1−1 +1−2 +1−2 1.111101100:+1.+0+0+1+1 +0+1 +0+1 +0+0 +0+0 +0−1 +0−1 +0−2 +0−2 1.111101101:+1.+0+0+1+1 −1+1 −1+1 −1+0 −1+0 −1−1 −1−1 −1−2 −1−2 1.111101110:+1.+0+0+1+0 +2+1 +2+1 +2+0 +2−1 +2−1 +2−2 +2−2 +1+1 1.111101111:+1.+0+0+1+0 +1+1 +1+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 1.111110000:+1.+0+0+1+0 +0+1 +0+0 +0+0 +0−1 +0−1 +0−2 +0−2 −1+1 1.111110001:+1.+0+0+1+0 −1+1 −1+0 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 1.111110010:+1.+0+0+1−1 +2+1 +2+0 +2−1 +2−1 +2−2 +2−2 +1+1 +1+1 1.111110011:+1.+0+0+1−1 +1+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 1.111110100:+1.+0+0+1−1 +0+0 +0+0 +0−1 +0−1 +0−2 +0−2 −1+1 −1+1 1.111110101:+1.+0+0+1−1 −1+0 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+1 1.111110110:+1.+0+0+1−2 +2+0 +2+0 +2−1 +2−1 +2−2 +2−2 +1+1 +1+1 1.111110111:+1.+0+0+1−2 +1+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 1.111111000:+1.+0+0+1−2 +0+0 +0−1 +0−1 +0−2 +0−2 −1+1 −1+1 −1+0 1.111111001:+1.+0+0+1−2 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+1 −2+0 1.111111010:+1.+0+0+0+1 +2+0 +2−1 +2−1 +2−2 +2−2 +1+1 +1+1 +1+0 1.111111011:+1.+0+0+0+1 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 +0+0 1.111111100:+1.+0+0+0+1 +0+0 +0−1 +0−1 +0−2 +0−2 −1+1 −1+1 −1+0 1.111111101:+1.+0+0+0+1 −1+0 −1−1 −1−1 −1−2 −1−2 −2+1 −2+1 −2+0 1.111111110:+1.+0+0+0+0 +2+0 +2−1 +2−1 +2−2 +2−2 +1+1 +1+1 +1+0 1.111111111:+1.+0+0+0+0 +1+0 +1−1 +1−1 +1−2 +1−2 +0+1 +0+1 +0+0

1. For use in a lookup table circuit of the type that is capable ofreceiving a plurality of bits that represent an input numeric value, andthat is capable of outputting a plurality of bits that represent anoutput numeric value that represents a result of operating on said inputnumeric value with a mathematical operation, a compressed direct lookuptable unit comprising: a redundant digits lookup table unit comprising aredundant digits lookup table, wherein said redundant digits lookuptable unit is capable of indexing said redundant digits lookup tableusing a first portion of said input bits of said input numeric value toselect a high order digits part of said output numeric value and aplurality of low order digits parts of said output numeric value; and alow order part multiplexer coupled to said redundant digits lookup tableunit, wherein said low order part multiplexer is capable of receivingsaid plurality of low order digits parts of said output numeric valuefrom said redundant digits lookup table unit, and is capable of using asecond portion of said input bits of said input numeric value to selecta low order digits part of said output numeric value from said pluralityof low order digit parts.
 2. A compressed direct lookup table unit asclaimed in claim 1 wherein said redundant digit output numeric valuecomprises a concatenation of said selected high order digits part fromsaid redundant digits lookup table unit and said selected low orderdigits part from said low order part multiplexer.
 3. A compressed directlookup table unit as claimed in claim 1 wherein said redundant digitslookup table does not create an output overflow exception condition. 4.A compressed direct lookup table unit as claimed in claim 1 wherein eachleading low order digit in said plurality of low order digits partsselected by said redundant digits lookup table unit has at most twosuccessive values within said plurality of low order digits parts.
 5. Acompressed direct lookup table unit as claimed in claim 1 wherein avalue of a leading low order digit in one of said plurality of low orderdigits parts selected by said redundant digits lookup table unit differsat most by one unit of value from a value of a leading low order digitin a low order digits part that is adjacent to said one of saidplurality of low order digits parts across a boundary between two linesof said lookup table.
 6. A compressed direct lookup table unit asclaimed in claim 1 wherein a size of said redundant digits lookup tableis smaller than a size of a standard binary direct lookup table.
 7. Acompressed direct lookup table unit as claimed in claim 1 wherein saidredundant digits lookup table comprises a plurality of table entrieshaving a radix two redundant digit format with digit values {−1, 0, 1}.8. A compressed direct lookup table unit as claimed in claim 1 whereinsaid redundant digits lookup table comprises a plurality of tableentries having a radix two redundant digit format with digit values {0,1, 2}.
 9. A compressed direct lookup table unit as claimed in claim 1wherein said redundant digits lookup table comprises a plurality oftable entries having a radix four redundant digit format with digitvalues {−2, −1, 0, 1, 2}.
 10. A compressed direct lockup table unit asclaimed in claim 1 wherein said redundant digits lookup table comprisesa plurality of table entries having a radix eight redundant digit formatwith digit values {−4, −3, −2, −1, 0, 1, 2, 3, 4}.
 11. A compresseddirect lookup table unit as claimed in claim 1 further comprising: amultiplier encoder coupled to an output of said redundant digits lookuptable unit and coupled to an output of said low order part multiplexer,said multiplier encoder capable of recoding said outputs into aredundant digit format capable of being directly used by a partialproduct generator of a Booth radix four multiplier.
 12. A compresseddirect lookup table unit as claimed in claim 1 wherein said redundantdigits lookup table comprises a “fourteen bits in and fourteen bits out”table capable of providing an output numeric value partitioned into ahigh order part of four “radix four” digits and a low order part ofthree “radix four” digits.
 13. A compressed direct lookup table unit asclaimed in claim 1 wherein said redundant digits lookup table comprisesa “sixteen bits in and sixteen bits out” table capable of providing anoutput numeric value partitioned into a high order part of five “radixfour” digits and a low order part of three “radix four” digits.
 14. Acompressed direct lookup table unit as claimed in claim 1 wherein saidredundant digits lookup table comprises a “eighteen bits in and eighteenbits out” table capable of providing an output numeric value partitionedinto a high order part of four “radix four” digits and a middle part oftwo “radix four” digits and a low order part of three “radix four”digits.
 15. For use in a lookup table circuit of the type that iscapable of receiving a plurality of bits that represent an input numericvalue, and that is capable of outputting a plurality of bits thatrepresent an output numeric value that represents a result of operatingon said input numeric value with a mathematical operation, a compresseddirect lookup table unit comprising: a redundant digits lookup tablerecoder comprising a redundant digits lookup table, wherein saidredundant digits lookup table recoder is capable of indexing saidredundant digits lookup table using a first portion of said input bitsof said input numeric value to select a high order digits part of saidoutput numeric value and a plurality of low order digits parts of saidoutput numeric value; wherein said redundant digits lookup table recoderis capable of recoding said high order digits part of said outputnumeric value to a redundant digit format and is capable of recodingsaid plurality of low order digits parts of said output numeric value toa redundant digit format; and a low order part multiplexer coupled tosaid redundant digits lookup table recoder, wherein said low order partmultiplexer is capable of receiving said recoded plurality of low orderdigits parts of said output numeric value from said redundant digitslookup table recoder, and is capable of using a second portion of saidinput bits of said input numeric value to select a recoded low orderdigits part of said output numeric value from said recoded plurality oflow order digit parts.
 16. A compressed direct lookup table unit asclaimed in claim 15 wherein said recoded redundant digit output numericvalue comprises a concatenation of said selected recoded high orderdigits part from said redundant digits lookup table recoder and saidselected recoded low order digits part from said low order partmultiplexer.
 17. A compressed direct lookup table unit as claimed inclaim 15 wherein said redundant digits lookup table does not create anoutput overflow exception condition.
 18. A compressed direct lookuptable unit as claimed in claim 15 wherein each leading low order digitin said plurality of low order digits parts selected by said redundantdigits lookup table unit has at most two successive values within saidplurality of low order digits parts.
 19. A compressed direct lookuptable unit as claimed in claim 15 wherein a value of a leading low orderdigit in one of said plurality of low order digits parts selected bysaid redundant digits lookup table unit differs at most by one unit ofvalue from a value of a leading low order digit in a low order digitspart that is adjacent to said one of said plurality of low order digitsparts across a boundary between two lines of said lookup table.
 20. Acompressed direct lookup table unit as claimed in claim 15 wherein asize of said redundant digits lookup table is smaller than a size of astandard binary direct lookup table.
 21. A compressed direct lookuptable unit as claimed in claim 15 wherein said redundant digits lookuptable comprises a plurality of table entries having radix four redundantdigit format with digit values {−2, −1, 0, 1, 2}.
 22. A compresseddirect lookup table unit as claimed in claim 21 wherein an outputnumeric value of said lookup table in said radix four redundant digitformat is capable of being used as an input to a partial productgenerator of a Booth radix four multiplier.
 23. A compressed directlookup table unit as claimed in claim 15 wherein said redundant digitslookup table comprises a plurality of table entries having a radix eightredundant digit format with digit values {−4, −3, −2, −1, 0, 1, 2, 3,4}.
 24. A compressed direct lookup table unit as claimed in claim 23wherein an output numeric value of said lookup table in said radix eightredundant digit format is capable of being used as an input to a partialproduct generator of a Booth radix eight multiplier.
 25. For use in alookup table circuit of the type that is capable of receiving aplurality of bits that represent an input numeric value, and that iscapable of outputting a plurality of bits that represent an outputnumeric value that represents a result of operating on said inputnumeric value with a mathematical operation, a compressed direct lookuptable unit comprising: a first redundant digits lookup table unitcomprising a first redundant digits lookup table, wherein said firstredundant digits lookup table unit is capable of indexing said firstredundant digits lookup table using a first portion of said input bitsof said input numeric value to select a first high order digits part ofa first output numeric value and a first plurality of low order digitsparts of said first output numeric value; a first low order partmultiplexer coupled to said first redundant digits lookup table unit,wherein said first low order part multiplexer is capable of receivingsaid first plurality of low order digits parts of said first outputnumeric value from said first redundant digits lookup table unit, and iscapable of using a second portion of said input bits of said inputnumeric value to select a first low order digits part of said outputnumeric value from said first plurality of low order digit parts; asecond redundant digits lookup table unit comprising a second redundantdigits lookup table, wherein said second redundant digits lookup tableunit is capable of indexing said second redundant digits lookup tableusing said first portion of said input bits of said input numeric valueto select a second high order digits part of a second output numericvalue and a second plurality of low order digits parts of said secondoutput numeric value; a second low order part multiplexer coupled tosaid second redundant digits lookup table unit, wherein said second loworder part multiplexer is capable of receiving said second plurality oflow order digits parts of said second output numeric value from saidsecond redundant digits lookup table unit, and is capable of using saidsecond portion of said input bits of said input numeric value to selecta second low order digits part of said second output numeric value fromsaid second plurality of low order digit parts; a read only memoryactivation unit coupled to said first and second redundant digits lookuptable units, said read only memory activation unit capable of activatingone of said first and second redundant digits lookup table units inresponse to receiving two leading bits of said input numeric value; anda multiplexer coupled to the output of said first and second redundantdigits table lookup units and coupled to the output of said first andsecond low order part multiplexers, said multiplexer capable ofoutputting one of said first output numeric value and said second outputnumeric value in response to receiving an activation signal from saidread only memory activation unit.
 26. For use in a lookup table circuitof the type that is capable of receiving a plurality of bits thatrepresent an input numeric value, and that is capable of outputting aplurality of bits that represent an output numeric value that representsa result of operating on said input numeric value with a mathematicaloperation, a method for looking up an output numeric value comprisingthe steps of: providing a redundant digits lookup table comprising afirst portion comprising a plurality of high order digits parts of saidoutput numeric value and a second portion comprising a plurality of loworder digits parts of said output numeric value; using a redundantdigits lookup table unit to index said redundant digits lookup tableusing a first portion of said input bits of said input numeric value toselect a high order digits part of said output numeric value and aplurality of low order digits parts of said output numeric value; andusing a low order part multiplexer to index said redundant digits lookuptable using a second portion of said input bits of said input numericvalue to select a low order digits part of said output numeric valuefrom said plurality of low order digits parts indexed by said redundantdigits lookup table unit.
 27. A method as claimed in claim 26 furthercomprising the step of: concatenating said selected high order digitspart from said redundant digits lookup table unit and said selected loworder digits part from said low order part multiplexer to form an outputnumeric value.
 28. A method as claimed in claim 26 further comprisingthe step of: selecting said low order digits part of said output numericvalue without creating an output overflow exception condition.
 29. Amethod as claimed in claim 26 wherein each leading low order digit insaid plurality of low order digits parts selected by said redundantdigits lookup table unit has at most two successive values within saidplurality of low order digits parts.
 30. A method as claimed in claim 26wherein a value of a leading low order digit in one of said plurality oflow order digits parts selected by said redundant digits lookup tableunit differs at most by one unit of value from a value of a leading loworder digit in a low order digits part that is adjacent to said one ofsaid plurality of low order digits parts across a boundary between twolines of said lookup table.
 31. A method as claimed in claim 26 whereinsaid output numeric value comprises one digit within a redundant digitvalue range and all other digits of said output numeric value are withina non-redundant digit value range.
 32. A method as claimed in claim 26wherein said one digit within said redundant digit value range occurs asa leading digit of said low order part of said output numeric value. 33.A method as claimed in claim 26 wherein said redundant digits lookuptable comprises a plurality of table entries having a radix tworedundant digit format with digit values {−1, 0, 1}.
 34. A method asclaimed in claim 26 wherein said redundant digits lookup table comprisesa plurality of table entries having a radix two redundant digit formatwith digit values {0, 1, 2}.
 35. A method as claimed in claim 26 whereinsaid redundant digits lookup table comprises a plurality of tableentries having a radix four redundant digit format with digit values{−2, −1, 0, 1, 2}.
 36. A method as claimed in claim 26 wherein saidredundant digits lookup table comprises a plurality of table entrieshaving a radix eight redundant digit format with digit values {−4, −3,−2, −1, 0, 1, 2, 3, 4}.
 37. A method as claimed in claim 35 furthercomprising the steps of: coupling a multiplier encoder to an output ofsaid redundant digits lookup table unit and to an output of said loworder part multiplexer; recoding said outputs into a redundant digitformat capable of being directly used by a partial product generator ofa Booth radix four multiplier.
 38. A method as claimed in claim 26wherein a size of said redundant digits table is smaller than a size ofa standard binary direct lookup table.
 39. A method as claimed in claim36 further comprising the steps of: coupling a multiplier encoder to anoutput of said redundant digits lookup table unit and to an output ofsaid low order part multiplexer; recoding said outputs into a redundantdigit format capable of being directly used by a partial productgenerator of a Booth radix eight multiplier.
 40. For use in a lookuptable circuit of the type that is capable of receiving a plurality ofbits that represent an input numeric value, and that is capable ofoutputting a plurality of bits that represent an output numeric valuethat represents a result of operating on said input numeric value with amathematical operation, a method for looking up an output numeric valuecomprising the steps of: providing a first redundant digits lookup tablecomprising a first portion comprising a first plurality of high orderdigits parts of a first output numeric value and a second portioncomprising a first plurality of low order digits parts of said firstoutput numeric value; using a redundant first digits lookup table unitto index said first redundant digits lookup table using a first portionof said input bits of said input numeric value to select a first highorder digits part of said first output numeric value and a firstplurality of low order digits parts of said output numeric value; andusing a first low order part multiplexer to index said first redundantdigits lookup table using a second portion of said input bits of saidinput numeric value to select a first low order digits part of saidfirst output numeric value from said first plurality of low order digitsparts indexed by said first redundant digits lookup table unit;providing a second redundant digits lookup table comprising a firstportion comprising a second plurality of high order digits parts of asecond output numeric value and a second portion comprising a secondplurality of low order digits parts of said second output numeric value;using a redundant second digits lookup table unit to index said secondredundant digits lookup table using said first portion of said inputbits of said input numeric value to select a second high order digitspart of said second output numeric value and a second plurality of loworder digits parts of said output numeric value; and using a second loworder part multiplexer to index said second redundant digits lookuptable using said second portion of said input bits of said input numericvalue to select a second low order digits part of said second outputnumeric value from said second plurality of low order digits partsindexed by said second redundant digits lookup table unit; activatingone of said first and second redundant digits lookup table units using aread only memory activation unit in response to receiving two leadingbits of said input numeric value in said read only memory activationunit; and outputting one of said first output numeric value and saidsecond output numeric value from a multiplexer in response to receivingin said multiplexer an activation signal from said read only memoryactivation unit.